Logic and Set Theory
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Operations on Sets
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Irrational Numbers
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Bounded and Unbounded Sets in R
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- Bounded set
- Supremum and Infimum
- Method for finding Supremum (L.U.B)
- Example (Finding Sup')
- Method for finding Infimum (G.L.B)
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4 - Density of Reals
- Exercise 5
- Exercise 6 - N is unbounded
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15 - Density of Rationals and Irrationals
- Exercise 16 - Dense Set, Rational and irrational
- Exercise 17
- Exercise 18

Further properties of bounded sets
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Mathematical Induction
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Summation and Sigma notation
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Famous Inequalities
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The Real Number System
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Logic and Set Theory","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Logical Connectives and Quantifiers","Duration":"4m 58s","ChapterTopicVideoID":25750,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"In this clip, we\u0027ll talk about logical connectives and quantifiers."},{"Start":"00:04.755 ","End":"00:08.730","Text":"In mathematics, we have various logical symbols"},{"Start":"00:08.730 ","End":"00:12.630","Text":"whose purpose is to enable mathematical precision and conciseness,"},{"Start":"00:12.630 ","End":"00:14.785","Text":"as well as having a pleasant appearance."},{"Start":"00:14.785 ","End":"00:18.305","Text":"Logical symbols or signs can be divided into 2 sets,"},{"Start":"00:18.305 ","End":"00:22.010","Text":"logical connectives and logical quantifiers."},{"Start":"00:22.010 ","End":"00:24.500","Text":"Let\u0027s start with the connectives."},{"Start":"00:24.500 ","End":"00:28.640","Text":"Logical connective is an object which either modifies a statement or combines"},{"Start":"00:28.640 ","End":"00:32.675","Text":"existing statements into a new statement called a compound statement."},{"Start":"00:32.675 ","End":"00:35.705","Text":"When we see the examples, all this will become clear."},{"Start":"00:35.705 ","End":"00:38.735","Text":"There are 5 main logical connectives."},{"Start":"00:38.735 ","End":"00:43.095","Text":"Negation, not, conjunction,"},{"Start":"00:43.095 ","End":"00:44.595","Text":"which is and,"},{"Start":"00:44.595 ","End":"00:47.910","Text":"and this is a symbol, disjunction, or,"},{"Start":"00:47.910 ","End":"00:49.315","Text":"with this symbol,"},{"Start":"00:49.315 ","End":"00:53.225","Text":"material implication, if something, then something,"},{"Start":"00:53.225 ","End":"00:55.105","Text":"and that\u0027s a right arrow,"},{"Start":"00:55.105 ","End":"00:56.915","Text":"and there\u0027s a biconditional,"},{"Start":"00:56.915 ","End":"00:59.395","Text":"if and only if, double arrow."},{"Start":"00:59.395 ","End":"01:02.195","Text":"Let\u0027s go into detail on each of these 5."},{"Start":"01:02.195 ","End":"01:05.150","Text":"Not A, it negates a statement."},{"Start":"01:05.150 ","End":"01:08.600","Text":"For example, if A is the statement x equals 2,"},{"Start":"01:08.600 ","End":"01:12.350","Text":"then not A is the statement x is not equal to 2."},{"Start":"01:12.350 ","End":"01:14.880","Text":"Next, A implies B,"},{"Start":"01:14.880 ","End":"01:17.415","Text":"or if A then B."},{"Start":"01:17.415 ","End":"01:20.740","Text":"Example, if x equals 2,"},{"Start":"01:20.740 ","End":"01:23.090","Text":"then x squared equals 4."},{"Start":"01:23.090 ","End":"01:25.910","Text":"Another example, the inverse statement,"},{"Start":"01:25.910 ","End":"01:29.480","Text":"x squared equals 4 implies x equals 2,"},{"Start":"01:29.480 ","End":"01:32.615","Text":"this implication is false."},{"Start":"01:32.615 ","End":"01:36.360","Text":"You can write it as does not imply."},{"Start":"01:36.360 ","End":"01:39.700","Text":"The line should be a bit more centered here, never mind."},{"Start":"01:39.700 ","End":"01:43.830","Text":"A if and only if B or A logically equivalent to B."},{"Start":"01:43.830 ","End":"01:47.555","Text":"An example, if x is not equal to 4,"},{"Start":"01:47.555 ","End":"01:53.455","Text":"that\u0027s equivalent to if and only if x is less than 4 or x is bigger than 4."},{"Start":"01:53.455 ","End":"01:58.790","Text":"A or B. Example, absolute value of x is bigger than"},{"Start":"01:58.790 ","End":"02:04.730","Text":"4 if and only if x is less than 4 or x is bigger than 4."},{"Start":"02:04.730 ","End":"02:07.075","Text":"It\u0027s the order we\u0027re stressing here."},{"Start":"02:07.075 ","End":"02:11.260","Text":"After all, we have A and B."},{"Start":"02:11.260 ","End":"02:16.100","Text":"As an example, absolute value of x is less than 4 if and only"},{"Start":"02:16.100 ","End":"02:21.110","Text":"if x is bigger than minus 4 and x is less than 4."},{"Start":"02:21.110 ","End":"02:26.330","Text":"Now, a remark, the following terms are synonymous: proposition,"},{"Start":"02:26.330 ","End":"02:30.110","Text":"statement, claim, at least in this course."},{"Start":"02:30.110 ","End":"02:33.190","Text":"I like claim because it\u0027s the shortest."},{"Start":"02:33.190 ","End":"02:35.390","Text":"Let\u0027s turn to logical quantifiers."},{"Start":"02:35.390 ","End":"02:39.200","Text":"We\u0027ve just spoken about logical connectives, now quantifiers."},{"Start":"02:39.200 ","End":"02:47.890","Text":"Many mathematical claims begin with for all x or to give you some examples,"},{"Start":"02:47.890 ","End":"02:52.430","Text":"there exists x such that x squared plus x equals 20."},{"Start":"02:52.430 ","End":"02:54.860","Text":"I\u0027m not concerned whether that\u0027s true or false,"},{"Start":"02:54.860 ","End":"02:56.480","Text":"just as a claim."},{"Start":"02:56.480 ","End":"03:02.704","Text":"Here\u0027s another claim, for all x there exists y such that x minus y equals 0."},{"Start":"03:02.704 ","End":"03:06.415","Text":"Here we have a for all and there exists."},{"Start":"03:06.415 ","End":"03:07.890","Text":"Here\u0027s another for all."},{"Start":"03:07.890 ","End":"03:09.800","Text":"For all natural numbers, n,"},{"Start":"03:09.800 ","End":"03:13.055","Text":"n squared minus n is an even number."},{"Start":"03:13.055 ","End":"03:15.350","Text":"It turns out that these 3 claims are all true,"},{"Start":"03:15.350 ","End":"03:17.030","Text":"but that\u0027s besides the point."},{"Start":"03:17.030 ","End":"03:22.740","Text":"Now, the expressions for all and there exists are called logical quantifiers,"},{"Start":"03:22.740 ","End":"03:24.605","Text":"and there are symbols for them."},{"Start":"03:24.605 ","End":"03:30.770","Text":"For all is an upside down A and there exists is a left to right"},{"Start":"03:30.770 ","End":"03:34.430","Text":"inverted E. These symbols let us write"},{"Start":"03:34.430 ","End":"03:39.130","Text":"things in more mathematical language and more concisely."},{"Start":"03:39.130 ","End":"03:42.200","Text":"For example, the above 3 claims,"},{"Start":"03:42.200 ","End":"03:44.425","Text":"we could write them as follows."},{"Start":"03:44.425 ","End":"03:49.865","Text":"First one, there exists x such that x squared plus x is 20."},{"Start":"03:49.865 ","End":"03:52.005","Text":"Here, for all x,"},{"Start":"03:52.005 ","End":"03:55.020","Text":"there exists y such that,"},{"Start":"03:55.020 ","End":"03:58.105","Text":"the colon is such that,"},{"Start":"03:58.105 ","End":"04:01.540","Text":"such that x minus y is 0."},{"Start":"04:01.540 ","End":"04:05.455","Text":"The last one, for all n,"},{"Start":"04:05.455 ","End":"04:10.900","Text":"there exists k such that n squared minus n is 2k,"},{"Start":"04:10.900 ","End":"04:13.615","Text":"that\u0027s how we indicate an even number."},{"Start":"04:13.615 ","End":"04:19.670","Text":"This is not so precise because I\u0027m implicitly assuming that n and k are natural numbers,"},{"Start":"04:19.670 ","End":"04:22.920","Text":"so here we can just write it more precisely."},{"Start":"04:22.920 ","End":"04:24.950","Text":"After the for all or there exists,"},{"Start":"04:24.950 ","End":"04:26.960","Text":"we can state where they come from,"},{"Start":"04:26.960 ","End":"04:28.070","Text":"which set they belong to."},{"Start":"04:28.070 ","End":"04:31.760","Text":"For all n in N, natural numbers,"},{"Start":"04:31.760 ","End":"04:37.630","Text":"there exists k in the set of natural numbers such that n squared minus n is 2k."},{"Start":"04:37.630 ","End":"04:40.310","Text":"Now an important remark,"},{"Start":"04:40.310 ","End":"04:44.060","Text":"you don\u0027t have to be able to use logical connectives."},{"Start":"04:44.060 ","End":"04:46.760","Text":"That\u0027s not expected in this course."},{"Start":"04:46.760 ","End":"04:51.050","Text":"What you\u0027re expected is just to be able to recognize and understand in"},{"Start":"04:51.050 ","End":"04:55.775","Text":"case you see them in the textbook or used by the teacher,"},{"Start":"04:55.775 ","End":"04:58.890","Text":"you should recognize and understand."}],"ID":26554},{"Watched":false,"Name":"Exercise 1","Duration":"4m 4s","ChapterTopicVideoID":25755,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"This exercise, we\u0027re given 5 claims and"},{"Start":"00:03.480 ","End":"00:06.600","Text":"we have to write them in words and then check if they\u0027re true or not."},{"Start":"00:06.600 ","End":"00:12.855","Text":"They all involve the for all and there exists the logical quantifiers."},{"Start":"00:12.855 ","End":"00:17.145","Text":"The first one, for all x and for all y,"},{"Start":"00:17.145 ","End":"00:21.300","Text":"we have x plus y squared is bigger than 0."},{"Start":"00:21.300 ","End":"00:24.420","Text":"That\u0027s in words, now, is it true or not?"},{"Start":"00:24.420 ","End":"00:26.790","Text":"Turns out that it\u0027s false,"},{"Start":"00:26.790 ","End":"00:29.565","Text":"and all we\u0027ll have to do is give 1 example."},{"Start":"00:29.565 ","End":"00:33.570","Text":"Take x equals 3 and y equals minus 3,"},{"Start":"00:33.570 ","End":"00:39.790","Text":"then x plus y is 0 and 0 squared is not bigger than 0, it\u0027s just equal."},{"Start":"00:39.790 ","End":"00:43.145","Text":"Now, number 2, for all x,"},{"Start":"00:43.145 ","End":"00:47.944","Text":"there exists y such that x plus y squared bigger than 0."},{"Start":"00:47.944 ","End":"00:49.895","Text":"This time, it is true."},{"Start":"00:49.895 ","End":"00:51.710","Text":"For example, whatever x is,"},{"Start":"00:51.710 ","End":"00:54.725","Text":"take y to be minus x plus 1,"},{"Start":"00:54.725 ","End":"00:57.860","Text":"and then x plus y will equal 1,"},{"Start":"00:57.860 ","End":"01:01.190","Text":"and 1 squared is certainly bigger than 0."},{"Start":"01:01.190 ","End":"01:02.610","Text":"Now, the next one."},{"Start":"01:02.610 ","End":"01:07.955","Text":"For all x and for all y, there exists z,"},{"Start":"01:07.955 ","End":"01:10.980","Text":"I say z like the Americans, if in England,"},{"Start":"01:10.980 ","End":"01:12.825","Text":"you would say zed of course,"},{"Start":"01:12.825 ","End":"01:17.355","Text":"there exists z such that xz equals y over 4."},{"Start":"01:17.355 ","End":"01:19.780","Text":"This turns out to be false,"},{"Start":"01:19.780 ","End":"01:22.730","Text":"and we just need 1 counterexample,"},{"Start":"01:22.730 ","End":"01:27.065","Text":"so take x equals 0 and take y equals 1."},{"Start":"01:27.065 ","End":"01:31.930","Text":"What we want is that 0 times z equals a quarter,"},{"Start":"01:31.930 ","End":"01:35.210","Text":"and that of course you can\u0027t have because 0 times anything is 0,"},{"Start":"01:35.210 ","End":"01:36.755","Text":"won\u0027t be a quarter."},{"Start":"01:36.755 ","End":"01:39.455","Text":"Next one, we have that,"},{"Start":"01:39.455 ","End":"01:44.240","Text":"for all x bigger than 0 and for all y bigger than 0,"},{"Start":"01:44.240 ","End":"01:47.620","Text":"or if you like, for all positive x and for all positive y,"},{"Start":"01:47.620 ","End":"01:48.910","Text":"it doesn\u0027t really matter."},{"Start":"01:48.910 ","End":"01:53.795","Text":"We have the square root of xy less than or equal to x plus y over 2."},{"Start":"01:53.795 ","End":"01:55.370","Text":"This is actually in words,"},{"Start":"01:55.370 ","End":"02:01.820","Text":"the geometric mean is less than or equal to the arithmetic mean of x and y."},{"Start":"02:01.820 ","End":"02:06.755","Text":"It turns out it\u0027s true and this is a famous theorem, let\u0027s prove it."},{"Start":"02:06.755 ","End":"02:09.380","Text":"Let\u0027s start with the following true statement,"},{"Start":"02:09.380 ","End":"02:12.110","Text":"x minus y squared is bigger or equal to 0,"},{"Start":"02:12.110 ","End":"02:13.940","Text":"whatever x and y are."},{"Start":"02:13.940 ","End":"02:16.565","Text":"Expanding that, we get the following."},{"Start":"02:16.565 ","End":"02:19.885","Text":"Now, let\u0027s add 4xy to both sides,"},{"Start":"02:19.885 ","End":"02:21.825","Text":"so this becomes plus 2xy,"},{"Start":"02:21.825 ","End":"02:23.575","Text":"and here we have 4xy."},{"Start":"02:23.575 ","End":"02:27.620","Text":"This we can write as x plus y squared bigger or equal to"},{"Start":"02:27.620 ","End":"02:32.080","Text":"4xy divided by 4 and then switch sides,"},{"Start":"02:32.080 ","End":"02:35.915","Text":"so xy is on this side and the x plus y squared over 4 on this side."},{"Start":"02:35.915 ","End":"02:38.200","Text":"Next, take the square root of both sides."},{"Start":"02:38.200 ","End":"02:39.530","Text":"x and y are positive,"},{"Start":"02:39.530 ","End":"02:41.770","Text":"so this is positive, everything is positive."},{"Start":"02:41.770 ","End":"02:45.620","Text":"The square root will just be x plus y over 2,"},{"Start":"02:45.620 ","End":"02:47.630","Text":"that\u0027s the positive 1."},{"Start":"02:47.630 ","End":"02:49.865","Text":"This is what we had to show."},{"Start":"02:49.865 ","End":"02:53.645","Text":"We have 1 more left to go back and look at it."},{"Start":"02:53.645 ","End":"03:02.225","Text":"For all n, there exists k such that n cubed minus n equals 6k,"},{"Start":"03:02.225 ","End":"03:05.650","Text":"n and k are natural numbers."},{"Start":"03:05.650 ","End":"03:11.960","Text":"Note, to say that something equals 6k is to say that it\u0027s divisible by 6."},{"Start":"03:11.960 ","End":"03:13.400","Text":"Let\u0027s write that."},{"Start":"03:13.400 ","End":"03:16.490","Text":"For all or for every natural number n,"},{"Start":"03:16.490 ","End":"03:19.850","Text":"n cubed minus n is divisible by 6."},{"Start":"03:19.850 ","End":"03:22.010","Text":"It turns out this is true."},{"Start":"03:22.010 ","End":"03:23.975","Text":"Let\u0027s prove it."},{"Start":"03:23.975 ","End":"03:25.710","Text":"N cubed minus n,"},{"Start":"03:25.710 ","End":"03:26.940","Text":"we can take n out,"},{"Start":"03:26.940 ","End":"03:29.520","Text":"n times n squared minus 1."},{"Start":"03:29.520 ","End":"03:34.260","Text":"n squared minus 1 is n minus 1 and plus 1, so it factorizes."},{"Start":"03:34.260 ","End":"03:36.980","Text":"What we see is that n cubed minus n is the product"},{"Start":"03:36.980 ","End":"03:39.425","Text":"of 3 consecutive numbers, just reorder them."},{"Start":"03:39.425 ","End":"03:42.480","Text":"It\u0027s n minus 1 times n times n plus 1,"},{"Start":"03:42.480 ","End":"03:44.495","Text":"these are consecutive numbers."},{"Start":"03:44.495 ","End":"03:46.700","Text":"So 1 of them at least is even,"},{"Start":"03:46.700 ","End":"03:48.700","Text":"could be 2 of them."},{"Start":"03:48.700 ","End":"03:52.440","Text":"Exactly, 1 of them is divisible by 3,"},{"Start":"03:52.440 ","End":"03:57.290","Text":"and when something is divisible by 2 and divisible by 3,"},{"Start":"03:57.290 ","End":"04:01.640","Text":"then it has to be divisible by 6 because 2 and 3 are relatively prime."},{"Start":"04:01.640 ","End":"04:05.280","Text":"That\u0027s what we had to show. We\u0027re done."}],"ID":26559},{"Watched":false,"Name":"Exercise 2","Duration":"1m 41s","ChapterTopicVideoID":25756,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this exercise, there are 4 parts and in each of them,"},{"Start":"00:03.180 ","End":"00:09.615","Text":"we have to translate what\u0027s written in words using logical signs,"},{"Start":"00:09.615 ","End":"00:13.065","Text":"which could be quantifiers or connectives."},{"Start":"00:13.065 ","End":"00:15.630","Text":"The first 1 says if x is bigger than 2,"},{"Start":"00:15.630 ","End":"00:17.645","Text":"then x squared is bigger than 4."},{"Start":"00:17.645 ","End":"00:19.345","Text":"This is an if-then,"},{"Start":"00:19.345 ","End":"00:22.410","Text":"so a right arrow, it will do it."},{"Start":"00:22.410 ","End":"00:25.470","Text":"In the next 1, you see that we have for all x,"},{"Start":"00:25.470 ","End":"00:28.395","Text":"so we\u0027re going to have to use a quantifier,"},{"Start":"00:28.395 ","End":"00:30.270","Text":"so for all x ,"},{"Start":"00:30.270 ","End":"00:34.600","Text":"then x squared plus 4 bigger than 0, just copy it."},{"Start":"00:34.600 ","End":"00:36.360","Text":"Next 1, for every,"},{"Start":"00:36.360 ","End":"00:37.695","Text":"it\u0027s like for all,"},{"Start":"00:37.695 ","End":"00:41.900","Text":"natural number n, n cubed minus n is divisible by 6."},{"Start":"00:41.900 ","End":"00:45.490","Text":"For all n and the divisible by 6,"},{"Start":"00:45.490 ","End":"00:49.490","Text":"we\u0027re going to use by saying that it\u0027s equal to 6k, so for all n,"},{"Start":"00:49.490 ","End":"00:55.200","Text":"there exists a k such that n cubed minus n equals 6k,"},{"Start":"00:55.200 ","End":"00:57.465","Text":"and we have to stipulate,"},{"Start":"00:57.465 ","End":"00:58.875","Text":"in case it\u0027s not understood,"},{"Start":"00:58.875 ","End":"01:01.235","Text":"that n and k are natural numbers."},{"Start":"01:01.235 ","End":"01:05.885","Text":"We could also write it like this: For all n in n,"},{"Start":"01:05.885 ","End":"01:07.620","Text":"there exists a k in n,"},{"Start":"01:07.620 ","End":"01:09.605","Text":"n is a set of natural numbers,"},{"Start":"01:09.605 ","End":"01:13.085","Text":"such that n cubed minus n equals 6k."},{"Start":"01:13.085 ","End":"01:15.320","Text":"Last 1, number 4,"},{"Start":"01:15.320 ","End":"01:17.390","Text":"the solution to the inequality,"},{"Start":"01:17.390 ","End":"01:19.805","Text":"absolute value of x less than 1,"},{"Start":"01:19.805 ","End":"01:23.225","Text":"is x between minus 1 and 1,"},{"Start":"01:23.225 ","End":"01:25.115","Text":"so that\u0027s an if and only if,"},{"Start":"01:25.115 ","End":"01:30.985","Text":"that x satisfies this if and only if x satisfies this,"},{"Start":"01:30.985 ","End":"01:33.080","Text":"so we have a double arrow,"},{"Start":"01:33.080 ","End":"01:41.730","Text":"absolute value of x less than 1 if and only if x between minus 1 and 1, and we\u0027re done."}],"ID":26560},{"Watched":false,"Name":"Sets and Elements","Duration":"4m 18s","ChapterTopicVideoID":25751,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"Next topic is basic set theory and these are the important concepts;"},{"Start":"00:04.770 ","End":"00:08.115","Text":"sets, elements, and the membership relation."},{"Start":"00:08.115 ","End":"00:12.985","Text":"Informally, a set in mathematics is a collection or set,"},{"Start":"00:12.985 ","End":"00:14.610","Text":"here it\u0027s a mathematical term,"},{"Start":"00:14.610 ","End":"00:16.230","Text":"here it\u0027s the real-world term."},{"Start":"00:16.230 ","End":"00:19.500","Text":"Collection or set, pretty much interchangeable of objects,"},{"Start":"00:19.500 ","End":"00:22.605","Text":"it doesn\u0027t have to be any relation between the objects."},{"Start":"00:22.605 ","End":"00:27.435","Text":"Each of the object is called an element or a member of the set."},{"Start":"00:27.435 ","End":"00:31.650","Text":"For example, just throw in 3 elements,"},{"Start":"00:31.650 ","End":"00:34.015","Text":"blue, Mars, and 7."},{"Start":"00:34.015 ","End":"00:37.339","Text":"They form a set, so no relation."},{"Start":"00:37.339 ","End":"00:39.755","Text":"Now, blue is a member of the set,"},{"Start":"00:39.755 ","End":"00:41.990","Text":"water is not a member of the set."},{"Start":"00:41.990 ","End":"00:47.690","Text":"Next thing is how do we describe the set? A simplest way is"},{"Start":"00:47.690 ","End":"00:53.630","Text":"just to list the members or elements in curly brackets separated by commas."},{"Start":"00:53.630 ","End":"00:55.220","Text":"In the example here,"},{"Start":"00:55.220 ","End":"00:56.855","Text":"what we get is blue,"},{"Start":"00:56.855 ","End":"01:01.210","Text":"Mars, 7 enclosed in curly braces."},{"Start":"01:01.210 ","End":"01:03.920","Text":"The set of numbers, for example,"},{"Start":"01:03.920 ","End":"01:05.180","Text":"containing 2, 3, 5,"},{"Start":"01:05.180 ","End":"01:08.510","Text":"and 7 can be denoted as curly braces,"},{"Start":"01:08.510 ","End":"01:11.440","Text":"2, 3, 5, 7."},{"Start":"01:11.440 ","End":"01:13.850","Text":"We often give sets a name,"},{"Start":"01:13.850 ","End":"01:18.610","Text":"typically an upper Latin letter like A."},{"Start":"01:18.610 ","End":"01:21.700","Text":"Could say A equals 2, 3, 5, 7."},{"Start":"01:21.700 ","End":"01:24.700","Text":"There are other ways of describing a set."},{"Start":"01:24.700 ","End":"01:29.210","Text":"They could be described in terms of a property that the elements have."},{"Start":"01:29.210 ","End":"01:32.810","Text":"We say the set of elements having property P,"},{"Start":"01:32.810 ","End":"01:35.810","Text":"where P is described in symbols and words."},{"Start":"01:35.810 ","End":"01:41.575","Text":"For example, we can say A is the set of integers between 1 and 20 inclusive."},{"Start":"01:41.575 ","End":"01:43.850","Text":"Using some mathematical symbols,"},{"Start":"01:43.850 ","End":"01:49.490","Text":"A equals the set of all x such that 1 is less than or equal to x,"},{"Start":"01:49.490 ","End":"01:52.750","Text":"less than or equal to 20, and x is an integer."},{"Start":"01:52.750 ","End":"01:55.550","Text":"We just say A is the set of x such that,"},{"Start":"01:55.550 ","End":"01:58.295","Text":"blah, blah, blah, any property."},{"Start":"01:58.295 ","End":"02:00.890","Text":"This is of a general form,"},{"Start":"02:00.890 ","End":"02:06.065","Text":"curly brace x and then a vertical line is such that,"},{"Start":"02:06.065 ","End":"02:10.670","Text":"and then some property of x where x satisfies property P. For example,"},{"Start":"02:10.670 ","End":"02:13.625","Text":"x is an integer between 1 and 20."},{"Start":"02:13.625 ","End":"02:18.545","Text":"Now in this case, when A is a finite set but not too many elements,"},{"Start":"02:18.545 ","End":"02:23.130","Text":"you could actually spell it out with curly braces and write A equals 1,"},{"Start":"02:23.130 ","End":"02:26.340","Text":"2, 3, 4, 5, I\u0027m not going to read it all out till 20."},{"Start":"02:26.340 ","End":"02:30.035","Text":"That\u0027s usually impractical, especially when you have a lot of elements."},{"Start":"02:30.035 ","End":"02:33.140","Text":"You can sometimes use the ellipsis saying 1, 2,"},{"Start":"02:33.140 ","End":"02:36.690","Text":"3,..., 20."},{"Start":"02:36.690 ","End":"02:38.705","Text":"Use that when the pattern is clear,"},{"Start":"02:38.705 ","End":"02:41.225","Text":"when you know how it behaves in the gap there."},{"Start":"02:41.225 ","End":"02:43.010","Text":"Let\u0027s take another example."},{"Start":"02:43.010 ","End":"02:47.035","Text":"I\u0027m going to give you several alternatives all to describe the same set A."},{"Start":"02:47.035 ","End":"02:52.670","Text":"We could say the set A is a set of solutions to the equation x squared minus 3,"},{"Start":"02:52.670 ","End":"02:54.245","Text":"x plus 2 equal 0."},{"Start":"02:54.245 ","End":"02:56.430","Text":"I\u0027m not going to solve it just yet."},{"Start":"02:56.430 ","End":"02:59.830","Text":"Now we could also say A equals set of all x,"},{"Start":"02:59.830 ","End":"03:04.945","Text":"such that x is a solution of x squared minus 3x plus 2 equal 0."},{"Start":"03:04.945 ","End":"03:07.780","Text":"Just slightly more mathematical."},{"Start":"03:07.780 ","End":"03:10.945","Text":"We could actually even drop the word as a solution of,"},{"Start":"03:10.945 ","End":"03:12.070","Text":"just say this,"},{"Start":"03:12.070 ","End":"03:16.330","Text":"A is the set of all x such that x squared minus 3x plus 2 equals 0."},{"Start":"03:16.330 ","End":"03:21.150","Text":"Another way you could do it is actually by solving this and saying that x could either be"},{"Start":"03:21.150 ","End":"03:26.645","Text":"1 or 2 so that A is the set containing 2 elements, 1, 2."},{"Start":"03:26.645 ","End":"03:29.420","Text":"Now let\u0027s talk about the membership relation."},{"Start":"03:29.420 ","End":"03:31.800","Text":"Membership of a set or in a set."},{"Start":"03:31.800 ","End":"03:34.575","Text":"Lets take the set 2, 3, 5,"},{"Start":"03:34.575 ","End":"03:38.565","Text":"7, and note that 3 is a member of this set."},{"Start":"03:38.565 ","End":"03:40.700","Text":"There\u0027s a mathematical way of writing that,"},{"Start":"03:40.700 ","End":"03:44.070","Text":"we write 3, this is E for element."},{"Start":"03:44.070 ","End":"03:48.515","Text":"Funny kind of E. 3 is an element of 2, 3, 5, 7,"},{"Start":"03:48.515 ","End":"03:53.910","Text":"or 3 is a member of or even 3 belongs to this set,"},{"Start":"03:53.910 ","End":"03:55.620","Text":"or 3 is in this set,"},{"Start":"03:55.620 ","End":"03:57.940","Text":"many ways of saying it."},{"Start":"03:58.040 ","End":"04:01.755","Text":"0 is not a member of this set,"},{"Start":"04:01.755 ","End":"04:04.850","Text":"so we write 0 is not an element of it,"},{"Start":"04:04.850 ","End":"04:06.030","Text":"it\u0027s not a symbol."},{"Start":"04:06.030 ","End":"04:08.750","Text":"In general, if a is an element of the set big A,"},{"Start":"04:08.750 ","End":"04:10.640","Text":"you write a belongs to A,"},{"Start":"04:10.640 ","End":"04:13.820","Text":"and if it\u0027s not you write a does not belong to A,"},{"Start":"04:13.820 ","End":"04:16.280","Text":"a is not an element of A."},{"Start":"04:16.280 ","End":"04:19.170","Text":"That\u0027s it for this clip."}],"ID":26555},{"Watched":false,"Name":"Exercise 3","Duration":"4m 32s","ChapterTopicVideoID":25757,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"This exercise is to test your understanding in"},{"Start":"00:04.725 ","End":"00:10.665","Text":"the relation member of and the relation subset of."},{"Start":"00:10.665 ","End":"00:12.480","Text":"We\u0027re given a set A,"},{"Start":"00:12.480 ","End":"00:13.650","Text":"which is the following,"},{"Start":"00:13.650 ","End":"00:18.210","Text":"and notice that it contains both numbers and sets of numbers."},{"Start":"00:18.210 ","End":"00:22.545","Text":"We have to mark each of the following as true or false."},{"Start":"00:22.545 ","End":"00:25.455","Text":"What I\u0027ll do is I\u0027ll just copy it,"},{"Start":"00:25.455 ","End":"00:26.985","Text":"make it easier,"},{"Start":"00:26.985 ","End":"00:30.390","Text":"and inside each rectangle,"},{"Start":"00:30.390 ","End":"00:32.490","Text":"I\u0027ll put a checkmark or a cross."},{"Start":"00:32.490 ","End":"00:33.750","Text":"I see the first 1."},{"Start":"00:33.750 ","End":"00:37.755","Text":"5 is a member or element of A,"},{"Start":"00:37.755 ","End":"00:39.720","Text":"and looking across here,"},{"Start":"00:39.720 ","End":"00:41.850","Text":"you might think you see a 5, well,"},{"Start":"00:41.850 ","End":"00:43.350","Text":"you do see a 5,"},{"Start":"00:43.350 ","End":"00:45.330","Text":"but the 5 is not in A,"},{"Start":"00:45.330 ","End":"00:47.720","Text":"the 5 is in this set,"},{"Start":"00:47.720 ","End":"00:49.655","Text":"and the set is a member of A,"},{"Start":"00:49.655 ","End":"00:50.945","Text":"but 5 isn\u0027t,"},{"Start":"00:50.945 ","End":"00:53.650","Text":"so the answer is no."},{"Start":"00:53.650 ","End":"00:56.730","Text":"Next, 2 is in A,"},{"Start":"00:56.730 ","End":"01:01.575","Text":"is a member of, yes because it\u0027s here, so yes."},{"Start":"01:01.575 ","End":"01:05.565","Text":"The set 2 is a member of A, yes."},{"Start":"01:05.565 ","End":"01:12.765","Text":"The set 2 is an element of A, so yes again."},{"Start":"01:12.765 ","End":"01:15.540","Text":"2 is a subset of A."},{"Start":"01:15.540 ","End":"01:18.710","Text":"Well, 2 is not even a set, so certainly no."},{"Start":"01:18.710 ","End":"01:22.235","Text":"A subset has to be a set."},{"Start":"01:22.235 ","End":"01:29.260","Text":"The set containing the element 2 is a subset of A."},{"Start":"01:29.260 ","End":"01:30.870","Text":"To be a subset,"},{"Start":"01:30.870 ","End":"01:33.545","Text":"every element here has to be an element here,"},{"Start":"01:33.545 ","End":"01:35.150","Text":"so let\u0027s take an element here."},{"Start":"01:35.150 ","End":"01:36.170","Text":"Well, there is only 1."},{"Start":"01:36.170 ","End":"01:40.145","Text":"The element here is the set containing 2, and is it in A?"},{"Start":"01:40.145 ","End":"01:41.845","Text":"Yes, it\u0027s here,"},{"Start":"01:41.845 ","End":"01:45.040","Text":"so yes. Next 1."},{"Start":"01:45.040 ","End":"01:48.980","Text":"Is the empty set a member of A?"},{"Start":"01:48.980 ","End":"01:50.495","Text":"Well, I\u0027m looking at the elements."},{"Start":"01:50.495 ","End":"01:51.680","Text":"Not this. It\u0027s not this."},{"Start":"01:51.680 ","End":"01:52.820","Text":"Not this. Not this."},{"Start":"01:52.820 ","End":"01:55.840","Text":"Not this. Not this, so no."},{"Start":"01:55.840 ","End":"01:58.815","Text":"The empty set is a subset of A."},{"Start":"01:58.815 ","End":"02:00.770","Text":"Without looking, I can say, yes,"},{"Start":"02:00.770 ","End":"02:03.810","Text":"the empty set is a subset of every set."},{"Start":"02:03.890 ","End":"02:11.090","Text":"The set containing 2 and the set containing 2 is a subset of A."},{"Start":"02:11.090 ","End":"02:15.590","Text":"We have to show that each element of this left-hand side is an element of A."},{"Start":"02:15.590 ","End":"02:17.705","Text":"There are 2 elements here. Let\u0027s check each 1."},{"Start":"02:17.705 ","End":"02:20.660","Text":"2, is that in A, a member?"},{"Start":"02:20.660 ","End":"02:24.185","Text":"Yes, we already see it here, and we see it here."},{"Start":"02:24.185 ","End":"02:26.375","Text":"The set containing 2,"},{"Start":"02:26.375 ","End":"02:28.460","Text":"that\u0027s here also,"},{"Start":"02:28.460 ","End":"02:30.155","Text":"so the answer is yes."},{"Start":"02:30.155 ","End":"02:32.000","Text":"The set containing 2,"},{"Start":"02:32.000 ","End":"02:35.130","Text":"4 is a subset of A."},{"Start":"02:35.130 ","End":"02:36.590","Text":"For this to be true,"},{"Start":"02:36.590 ","End":"02:39.470","Text":"we have to have each element that\u0027s here,"},{"Start":"02:39.470 ","End":"02:40.820","Text":"also an element here,"},{"Start":"02:40.820 ","End":"02:43.510","Text":"2 is in a, yes."},{"Start":"02:43.510 ","End":"02:45.805","Text":"4 is a member of A?"},{"Start":"02:45.805 ","End":"02:48.439","Text":"Yes, so yes."},{"Start":"02:48.439 ","End":"02:53.010","Text":"4, 2 is a member of A."},{"Start":"02:53.010 ","End":"02:54.930","Text":"Well, 4, 2 is the same as 2,"},{"Start":"02:54.930 ","End":"02:56.670","Text":"4, and 2,"},{"Start":"02:56.670 ","End":"02:59.580","Text":"4 is also a member of A,"},{"Start":"02:59.580 ","End":"03:00.960","Text":"we see it here,"},{"Start":"03:00.960 ","End":"03:02.650","Text":"so this is peculiar."},{"Start":"03:02.650 ","End":"03:07.220","Text":"2, 4 is both a subset of A and a member of A, that\u0027s okay."},{"Start":"03:07.220 ","End":"03:09.710","Text":"The set containing the set 2,"},{"Start":"03:09.710 ","End":"03:11.525","Text":"4 is a member of A."},{"Start":"03:11.525 ","End":"03:13.040","Text":"Well, no,"},{"Start":"03:13.040 ","End":"03:17.595","Text":"because if you go across the members of A,"},{"Start":"03:17.595 ","End":"03:19.080","Text":"how many are there, by the way?"},{"Start":"03:19.080 ","End":"03:21.505","Text":"1, 2, 3, 4, 5, 6."},{"Start":"03:21.505 ","End":"03:24.695","Text":"None of them is equal to this."},{"Start":"03:24.695 ","End":"03:27.290","Text":"This is actually a subset of A,"},{"Start":"03:27.290 ","End":"03:30.925","Text":"but it\u0027s not a member of A, so no."},{"Start":"03:30.925 ","End":"03:32.970","Text":"The set containing 2,"},{"Start":"03:32.970 ","End":"03:36.660","Text":"5, is that a subset of A?"},{"Start":"03:36.660 ","End":"03:38.760","Text":"For that to be true, 2 has to be in A,"},{"Start":"03:38.760 ","End":"03:41.100","Text":"and 5 has to be in A as members."},{"Start":"03:41.100 ","End":"03:46.620","Text":"2 is, but 5 isn\u0027t, so no."},{"Start":"03:46.620 ","End":"03:49.440","Text":"Is the same thing an element of A?"},{"Start":"03:49.440 ","End":"03:50.760","Text":"Well, yes,"},{"Start":"03:50.760 ","End":"03:54.480","Text":"because here it is, so yes."},{"Start":"03:54.480 ","End":"03:56.880","Text":"The set containing 1, 4,"},{"Start":"03:56.880 ","End":"03:59.220","Text":"is that an element of A? Well, no."},{"Start":"03:59.220 ","End":"04:01.200","Text":"It\u0027s not this 1. It\u0027s not this 1."},{"Start":"04:01.200 ","End":"04:03.330","Text":"It\u0027s not this set, so no."},{"Start":"04:03.330 ","End":"04:06.790","Text":"The set containing 2 is a subset of A."},{"Start":"04:06.790 ","End":"04:09.560","Text":"This will be true if every element here is an element here."},{"Start":"04:09.560 ","End":"04:11.030","Text":"The element here is 2,"},{"Start":"04:11.030 ","End":"04:14.500","Text":"and 2 does belong to A, this 2."},{"Start":"04:14.500 ","End":"04:17.490","Text":"The set containing the set 2,"},{"Start":"04:17.490 ","End":"04:19.200","Text":"4 is a subset of A?"},{"Start":"04:19.200 ","End":"04:22.160","Text":"For this to be true, every element here has to be an element here."},{"Start":"04:22.160 ","End":"04:23.900","Text":"The only element is the set 2,"},{"Start":"04:23.900 ","End":"04:25.505","Text":"4, and yes,"},{"Start":"04:25.505 ","End":"04:28.280","Text":"it is indeed an element of A,"},{"Start":"04:28.280 ","End":"04:29.735","Text":"so this is a subset,"},{"Start":"04:29.735 ","End":"04:32.790","Text":"yes, and we\u0027re done."}],"ID":26561},{"Watched":false,"Name":"Equality of Sets","Duration":"2m 40s","ChapterTopicVideoID":25754,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.329","Text":"Now we\u0027ll talk about equality of sets."},{"Start":"00:03.329 ","End":"00:07.950","Text":"Sets are completely characterized by their members."},{"Start":"00:07.950 ","End":"00:14.565","Text":"2 sets are equal if and only if they have exactly the same members or elements."},{"Start":"00:14.565 ","End":"00:18.300","Text":"For example, look at these 2 sets."},{"Start":"00:18.300 ","End":"00:21.795","Text":"They appear different, but they\u0027re not."},{"Start":"00:21.795 ","End":"00:26.490","Text":"They are the same because every element of 1 is an element of the other,"},{"Start":"00:26.490 ","End":"00:29.715","Text":"if you say in the second one and 5 is an element,"},{"Start":"00:29.715 ","End":"00:31.365","Text":"well, 5 is an element here."},{"Start":"00:31.365 ","End":"00:33.780","Text":"3 is an element here, 3 is also an element here."},{"Start":"00:33.780 ","End":"00:38.620","Text":"Everything that you could point that\u0027s an element here is also an element here."},{"Start":"00:38.620 ","End":"00:43.069","Text":"As you can see, the way we order them in curly braces,"},{"Start":"00:43.069 ","End":"00:45.980","Text":"we could list them multiple times."},{"Start":"00:45.980 ","End":"00:47.480","Text":"Give an element, like here,"},{"Start":"00:47.480 ","End":"00:49.990","Text":"5 and 5 is repeated,"},{"Start":"00:49.990 ","End":"00:51.675","Text":"and 7 and 7."},{"Start":"00:51.675 ","End":"00:52.950","Text":"Also the order."},{"Start":"00:52.950 ","End":"00:55.560","Text":"Here we have 3 then 5,"},{"Start":"00:55.560 ","End":"00:57.720","Text":"here 5 twice and then 3."},{"Start":"00:57.720 ","End":"00:59.405","Text":"Doesn\u0027t make any difference."},{"Start":"00:59.405 ","End":"01:03.050","Text":"It\u0027s just characterized by which members it has."},{"Start":"01:03.050 ","End":"01:04.620","Text":"Just repeating myself."},{"Start":"01:04.620 ","End":"01:07.485","Text":"All that matters inequalities is that 2 sets have the same members."},{"Start":"01:07.485 ","End":"01:09.170","Text":"Now we can define it formally."},{"Start":"01:09.170 ","End":"01:13.575","Text":"Set A is equal to set B is equivalent to saying that"},{"Start":"01:13.575 ","End":"01:18.045","Text":"element x belongs to A if and only if x belongs to B."},{"Start":"01:18.045 ","End":"01:21.695","Text":"Or x is a member of A if and only if x is a member of B."},{"Start":"01:21.695 ","End":"01:27.300","Text":"Membership in the 2 sets is equivalent and then we said the sets are equal."},{"Start":"01:27.300 ","End":"01:29.280","Text":"Now, let\u0027s do an exercise."},{"Start":"01:29.280 ","End":"01:33.430","Text":"In this exercise, we\u0027re given 4 descriptions of sets A, B, C,"},{"Start":"01:33.430 ","End":"01:39.684","Text":"D. We have to decide which of these are equal to each other and which are different."},{"Start":"01:39.684 ","End":"01:42.085","Text":"Let\u0027s go over them one by one."},{"Start":"01:42.085 ","End":"01:46.150","Text":"What we\u0027ll do is we\u0027ll write out the elements of each"},{"Start":"01:46.150 ","End":"01:50.215","Text":"without repetition in ascending order so we can see what\u0027s the same and what\u0027s different."},{"Start":"01:50.215 ","End":"01:54.310","Text":"The first one as is 2, 5, 7."},{"Start":"01:54.310 ","End":"01:57.550","Text":"The next one, if you solve this equation,"},{"Start":"01:57.550 ","End":"02:00.265","Text":"we get x equals 2 or x equals 5."},{"Start":"02:00.265 ","End":"02:04.120","Text":"The set of all such x is just the set 2, 5."},{"Start":"02:04.120 ","End":"02:07.785","Text":"The third set, is the set of all primes,"},{"Start":"02:07.785 ","End":"02:12.465","Text":"whole numbers between 2 and 10 and turns out there\u0027s 3 of them,"},{"Start":"02:12.465 ","End":"02:14.535","Text":"2, 5, and 7."},{"Start":"02:14.535 ","End":"02:19.220","Text":"The set d if you throw out duplicates and rearrange,"},{"Start":"02:19.220 ","End":"02:21.625","Text":"we also get 2, 5, 7."},{"Start":"02:21.625 ","End":"02:23.980","Text":"We see from this that A, C,"},{"Start":"02:23.980 ","End":"02:29.040","Text":"and D are equal to each other and B is different,"},{"Start":"02:29.040 ","End":"02:32.370","Text":"so B different from A because, for example,"},{"Start":"02:32.370 ","End":"02:35.070","Text":"7 belongs to A and 7 doesn\u0027t belong to B,"},{"Start":"02:35.070 ","End":"02:37.565","Text":"so it\u0027s not the same by definition."},{"Start":"02:37.565 ","End":"02:41.010","Text":"With that, we conclude this clip."}],"ID":26558},{"Watched":false,"Name":"Exercise 4","Duration":"1m 31s","ChapterTopicVideoID":25758,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this exercise, we\u0027re given 5 descriptions of"},{"Start":"00:04.140 ","End":"00:08.265","Text":"sets and we have to find which are equal to 1 another."},{"Start":"00:08.265 ","End":"00:12.495","Text":"What we\u0027ll do is we\u0027ll rewrite them into the same form,"},{"Start":"00:12.495 ","End":"00:14.790","Text":"like in curly braces."},{"Start":"00:14.790 ","End":"00:18.030","Text":"Then we can see which are equal and which are not."},{"Start":"00:18.030 ","End":"00:21.495","Text":"The first 1 take as is,"},{"Start":"00:21.495 ","End":"00:23.100","Text":"the second 1,"},{"Start":"00:23.100 ","End":"00:26.085","Text":"set of natural numbers between 10 and 20,"},{"Start":"00:26.085 ","End":"00:27.945","Text":"which are prime numbers."},{"Start":"00:27.945 ","End":"00:29.595","Text":"If we compute that,"},{"Start":"00:29.595 ","End":"00:31.910","Text":"then it comes out to be the same."},{"Start":"00:31.910 ","End":"00:35.065","Text":"These are the prime numbers between 10 and 20."},{"Start":"00:35.065 ","End":"00:37.440","Text":"Then we have the following set."},{"Start":"00:37.440 ","End":"00:40.740","Text":"Let\u0027s just try and duplicate because that makes no difference."},{"Start":"00:40.740 ","End":"00:42.480","Text":"Again, we have this."},{"Start":"00:42.480 ","End":"00:44.160","Text":"Already we see that A,"},{"Start":"00:44.160 ","End":"00:45.840","Text":"B, and C are equal."},{"Start":"00:45.840 ","End":"00:47.535","Text":"Now what about D?"},{"Start":"00:47.535 ","End":"00:52.100","Text":"The integers which are equal to 4 times another integer,"},{"Start":"00:52.100 ","End":"00:55.385","Text":"which means integers divisible by 4."},{"Start":"00:55.385 ","End":"00:58.460","Text":"That\u0027s 0, 4, 8, 12, etc,"},{"Start":"00:58.460 ","End":"01:01.250","Text":"and minus 4, minus 8, minus 12."},{"Start":"01:01.250 ","End":"01:04.175","Text":"This is the list of members."},{"Start":"01:04.175 ","End":"01:06.830","Text":"It\u0027s infinite, but we see the pattern."},{"Start":"01:06.830 ","End":"01:09.725","Text":"In E, x is an integer,"},{"Start":"01:09.725 ","End":"01:12.920","Text":"but it\u0027s equal to twice an even number."},{"Start":"01:12.920 ","End":"01:18.490","Text":"Now, twice an even number is exactly the same as being divisible by 4."},{"Start":"01:18.490 ","End":"01:21.405","Text":"Once again, we get this."},{"Start":"01:21.405 ","End":"01:24.160","Text":"D and E are equal also."},{"Start":"01:24.160 ","End":"01:26.340","Text":"To summarize, A, B,"},{"Start":"01:26.340 ","End":"01:27.945","Text":"and C are equal,"},{"Start":"01:27.945 ","End":"01:32.440","Text":"and D and E are equal. We\u0027re done."}],"ID":26562},{"Watched":false,"Name":"Finits, Infinite and Empty Sets","Duration":"3m 19s","ChapterTopicVideoID":25748,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.990","Text":"Continuing with sets, we can classify sets based on their size, broad categories."},{"Start":"00:06.990 ","End":"00:08.340","Text":"There are finite sets,"},{"Start":"00:08.340 ","End":"00:09.510","Text":"there are infinite sets,"},{"Start":"00:09.510 ","End":"00:11.475","Text":"and there\u0027s an empty set."},{"Start":"00:11.475 ","End":"00:13.200","Text":"It\u0027s very straightforward."},{"Start":"00:13.200 ","End":"00:16.920","Text":"A set with a finite number of elements is called a finite set,"},{"Start":"00:16.920 ","End":"00:21.210","Text":"and a set with an infinite number of elements is called an infinite set."},{"Start":"00:21.210 ","End":"00:23.100","Text":"There\u0027s a special case,"},{"Start":"00:23.100 ","End":"00:26.800","Text":"a set with no elements factor is only 1 set with no elements,"},{"Start":"00:26.800 ","End":"00:33.055","Text":"it\u0027s unique and it\u0027s called the empty set and I guess we\u0027d consider it as a finite set."},{"Start":"00:33.055 ","End":"00:35.105","Text":"Here are some examples."},{"Start":"00:35.105 ","End":"00:37.400","Text":"The set 2, 3, 5,"},{"Start":"00:37.400 ","End":"00:39.560","Text":"7 is a finite set,"},{"Start":"00:39.560 ","End":"00:41.240","Text":"but the set 1, 2, 3,"},{"Start":"00:41.240 ","End":"00:44.720","Text":"4,... is an infinite set,"},{"Start":"00:44.720 ","End":"00:47.780","Text":"the set of positive natural numbers."},{"Start":"00:47.780 ","End":"00:51.080","Text":"We can talk about the cardinality of a set,"},{"Start":"00:51.080 ","End":"00:53.900","Text":"that is the number of elements in a set."},{"Start":"00:53.900 ","End":"00:56.360","Text":"In the case of a finite set,"},{"Start":"00:56.360 ","End":"01:02.705","Text":"the number of elements is the cardinality and we denote that as A in bass,"},{"Start":"01:02.705 ","End":"01:04.700","Text":"like an absolute value."},{"Start":"01:04.700 ","End":"01:06.920","Text":"In the example above,"},{"Start":"01:06.920 ","End":"01:10.990","Text":"the cardinality of B is 4."},{"Start":"01:10.990 ","End":"01:13.865","Text":"Number of elements in an infinite set,"},{"Start":"01:13.865 ","End":"01:18.350","Text":"we just write infinity so that the cardinality of C is"},{"Start":"01:18.350 ","End":"01:24.440","Text":"infinity and number of elements in the empty set is 0."},{"Start":"01:24.440 ","End":"01:27.785","Text":"What I said here is not quite precise."},{"Start":"01:27.785 ","End":"01:30.710","Text":"There are actually different kinds of infinity,"},{"Start":"01:30.710 ","End":"01:33.925","Text":"but that\u0027s beyond the scope of this course."},{"Start":"01:33.925 ","End":"01:35.580","Text":"You take any 2 infinities,"},{"Start":"01:35.580 ","End":"01:39.160","Text":"1 will be bigger or smaller or equal to the other."},{"Start":"01:39.160 ","End":"01:42.140","Text":"There are larger infinities and smaller infinities."},{"Start":"01:42.140 ","End":"01:46.295","Text":"For example, the size of the rational numbers"},{"Start":"01:46.295 ","End":"01:51.090","Text":"is less than the size of the set of real numbers."},{"Start":"01:51.090 ","End":"01:55.205","Text":"The natural numbers and the rational numbers have the same kind of infinity."},{"Start":"01:55.205 ","End":"01:57.395","Text":"But like I said, it\u0027s beyond the scope."},{"Start":"01:57.395 ","End":"02:00.185","Text":"Let\u0027s say a few words about the empty set."},{"Start":"02:00.185 ","End":"02:04.305","Text":"There\u0027s only 1 empty set and it has no elements."},{"Start":"02:04.305 ","End":"02:08.945","Text":"It\u0027s called the empty set and is denoted by the Greek letter Phi,"},{"Start":"02:08.945 ","End":"02:13.835","Text":"and sometimes just by curly braces with nothing inside them."},{"Start":"02:13.835 ","End":"02:17.465","Text":"For example, the following sets are actually"},{"Start":"02:17.465 ","End":"02:22.520","Text":"the empty set and are equal to Phi set of all unicorns."},{"Start":"02:22.520 ","End":"02:25.490","Text":"Maybe some people believe that it\u0027s not empty,"},{"Start":"02:25.490 ","End":"02:28.070","Text":"but generally considered to be an empty set."},{"Start":"02:28.070 ","End":"02:31.160","Text":"Set the goal people over 200 years old."},{"Start":"02:31.160 ","End":"02:35.975","Text":"Again, I\u0027m not going to get into factual arguments that you can say it\u0027s an empty set,"},{"Start":"02:35.975 ","End":"02:39.230","Text":"set of all negative numbers bigger than 5,"},{"Start":"02:39.230 ","End":"02:48.905","Text":"that\u0027s for sure the empty set and set of all x such that 0 times x is 4, also empty."},{"Start":"02:48.905 ","End":"02:51.055","Text":"So there is a some examples."},{"Start":"02:51.055 ","End":"02:53.235","Text":"Now a warning,"},{"Start":"02:53.235 ","End":"02:56.315","Text":"some people make the mistake of writing the empty set"},{"Start":"02:56.315 ","End":"02:59.570","Text":"as 0 in curly braces and it\u0027s wrong."},{"Start":"02:59.570 ","End":"03:02.675","Text":"I don\u0027t know why some people do that, just beware."},{"Start":"03:02.675 ","End":"03:04.640","Text":"The empty set has no elements,"},{"Start":"03:04.640 ","End":"03:06.755","Text":"but this set has 1 element."},{"Start":"03:06.755 ","End":"03:08.090","Text":"The element 0 is in it,"},{"Start":"03:08.090 ","End":"03:10.100","Text":"so empty set and zero,"},{"Start":"03:10.100 ","End":"03:11.390","Text":"not the same thing."},{"Start":"03:11.390 ","End":"03:14.510","Text":"In fact, the cardinality of the empty set is 0,"},{"Start":"03:14.510 ","End":"03:17.225","Text":"but the cardinality of this is 1."},{"Start":"03:17.225 ","End":"03:20.310","Text":"That concludes this clip."}],"ID":26552},{"Watched":false,"Name":"Subsets","Duration":"2m 50s","ChapterTopicVideoID":25753,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"Now a new concept,"},{"Start":"00:02.010 ","End":"00:03.960","Text":"the concept of a subset."},{"Start":"00:03.960 ","End":"00:06.155","Text":"Consider the following 2 sets."},{"Start":"00:06.155 ","End":"00:07.975","Text":"A, which contains 1 and 2,"},{"Start":"00:07.975 ","End":"00:10.410","Text":"and B contains 1, 2, 3, and 4."},{"Start":"00:10.410 ","End":"00:15.060","Text":"Now, every element in A is also an element of B."},{"Start":"00:15.060 ","End":"00:16.710","Text":"The 1 is in B,"},{"Start":"00:16.710 ","End":"00:18.360","Text":"the 2 is in B."},{"Start":"00:18.360 ","End":"00:22.125","Text":"In that case, you say that A is a subset of B,"},{"Start":"00:22.125 ","End":"00:24.990","Text":"or that A is contained in B."},{"Start":"00:24.990 ","End":"00:28.725","Text":"The notation for that is this symbol,"},{"Start":"00:28.725 ","End":"00:30.705","Text":"A is a subset of B."},{"Start":"00:30.705 ","End":"00:32.370","Text":"In our case, set 1,"},{"Start":"00:32.370 ","End":"00:35.280","Text":"2 is a subset of the set 1, 2, 3, 4."},{"Start":"00:35.280 ","End":"00:38.385","Text":"There\u0027s a logical definition of subset."},{"Start":"00:38.385 ","End":"00:44.350","Text":"To say that A is a subset of B is equivalent to saying that if x is in A,"},{"Start":"00:44.350 ","End":"00:46.600","Text":"then x is in B always,"},{"Start":"00:46.600 ","End":"00:47.800","Text":"this is an implication."},{"Start":"00:47.800 ","End":"00:49.060","Text":"Whenever x is in A,"},{"Start":"00:49.060 ","End":"00:50.650","Text":"x is in B."},{"Start":"00:50.650 ","End":"00:54.040","Text":"In our example, A is a subset of B,"},{"Start":"00:54.040 ","End":"00:57.910","Text":"but B is not a subset of A because for example,"},{"Start":"00:57.910 ","End":"00:59.815","Text":"3 belongs to B."},{"Start":"00:59.815 ","End":"01:01.570","Text":"3 is an element of B,"},{"Start":"01:01.570 ","End":"01:04.295","Text":"but 3 is not an element of A."},{"Start":"01:04.295 ","End":"01:06.975","Text":"This implication wouldn\u0027t work here."},{"Start":"01:06.975 ","End":"01:10.660","Text":"We would write that B is not a subset of A,"},{"Start":"01:10.660 ","End":"01:12.325","Text":"subset with a line through it."},{"Start":"01:12.325 ","End":"01:14.830","Text":"Another example, take C as 1, 2,"},{"Start":"01:14.830 ","End":"01:16.540","Text":"5 and D is 1,"},{"Start":"01:16.540 ","End":"01:19.360","Text":"4, neither is contained in the other."},{"Start":"01:19.360 ","End":"01:24.680","Text":"C is not a subset of D because of the 5 or the 2."},{"Start":"01:24.680 ","End":"01:30.415","Text":"Let\u0027s say 5 is in C but not in D. The other way around is also not a subset,"},{"Start":"01:30.415 ","End":"01:31.835","Text":"you\u0027d have to choose the 4,"},{"Start":"01:31.835 ","End":"01:34.865","Text":"4 is in D but 4 is not in C,"},{"Start":"01:34.865 ","End":"01:37.250","Text":"neither is contained in the other."},{"Start":"01:37.250 ","End":"01:43.595","Text":"Note that any set A has itself and the empty set of subsets."},{"Start":"01:43.595 ","End":"01:48.910","Text":"A is a subset of A and the empty set is a subset of A."},{"Start":"01:48.910 ","End":"01:55.010","Text":"Remark, do you want to indicate that A is a proper subset of B,"},{"Start":"01:55.010 ","End":"01:59.105","Text":"proper meaning contained in but not equal to."},{"Start":"01:59.105 ","End":"02:04.795","Text":"Then you write A subset of B like this without the line here."},{"Start":"02:04.795 ","End":"02:09.290","Text":"For example, the set containing 1 and 2 is a proper subset of 1,"},{"Start":"02:09.290 ","End":"02:12.650","Text":"2, 3, 4 because it\u0027s a subset and they not equal."},{"Start":"02:12.650 ","End":"02:15.530","Text":"Now let\u0027s revisit the equality of sets."},{"Start":"02:15.530 ","End":"02:21.440","Text":"2 sets are equal if and only if each is contained in the other."},{"Start":"02:21.440 ","End":"02:24.860","Text":"We already had that A equals B is"},{"Start":"02:24.860 ","End":"02:28.250","Text":"equivalent to x belongs to A if and only if x belongs to B."},{"Start":"02:28.250 ","End":"02:33.620","Text":"Now, the if and only if can be interpreted as if and only if,"},{"Start":"02:33.620 ","End":"02:36.349","Text":"if you take the arrow facing to the right,"},{"Start":"02:36.349 ","End":"02:38.885","Text":"then it says that A is a subset of B."},{"Start":"02:38.885 ","End":"02:40.430","Text":"If you take the arrow to the left,"},{"Start":"02:40.430 ","End":"02:42.725","Text":"it says that B is a subset of A."},{"Start":"02:42.725 ","End":"02:48.860","Text":"A equals B if and only if A is a subset of B and B is a subset of A."},{"Start":"02:48.860 ","End":"02:51.720","Text":"That concludes this clip."}],"ID":26557},{"Watched":false,"Name":"Special Sets of Numbers","Duration":"7m 31s","ChapterTopicVideoID":25752,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.365","Text":"In this clip, we\u0027ll talk about some special sets of numbers,"},{"Start":"00:04.365 ","End":"00:09.810","Text":"and these are sets which are very important and they\u0027re used very frequently,"},{"Start":"00:09.810 ","End":"00:13.810","Text":"and so they\u0027ve been given special names and notations."},{"Start":"00:13.810 ","End":"00:17.885","Text":"Let\u0027s start with the set of natural numbers that you\u0027re all familiar with,"},{"Start":"00:17.885 ","End":"00:19.925","Text":"like 1, 2, 3, 4, etc."},{"Start":"00:19.925 ","End":"00:24.720","Text":"The set is used both for counting and that\u0027s what we call cardinal numbers,"},{"Start":"00:24.720 ","End":"00:28.310","Text":"and for ordering, these would be the ordinal numbers."},{"Start":"00:28.310 ","End":"00:30.215","Text":"In the English language,"},{"Start":"00:30.215 ","End":"00:33.170","Text":"there are the terms cardinal numbers, 1, 2, 3,"},{"Start":"00:33.170 ","End":"00:36.410","Text":"4, etc, and there are ordinal numbers 1st,"},{"Start":"00:36.410 ","End":"00:38.015","Text":"2nd, 3rd, 4th."},{"Start":"00:38.015 ","End":"00:40.210","Text":"Let\u0027s gives some examples."},{"Start":"00:40.210 ","End":"00:43.370","Text":"Examples of used for counting,"},{"Start":"00:43.370 ","End":"00:45.784","Text":"say there are 4 diners at the table."},{"Start":"00:45.784 ","End":"00:50.360","Text":"Joe has $5 and ordering examples,"},{"Start":"00:50.360 ","End":"00:53.435","Text":"Birmingham is the second largest city in the UK,"},{"Start":"00:53.435 ","End":"00:56.135","Text":"maybe it is, maybe it isn\u0027t but that\u0027s precisely the point."},{"Start":"00:56.135 ","End":"00:58.990","Text":"Dan is seventh in line in the movie queue."},{"Start":"00:58.990 ","End":"01:01.759","Text":"There\u0027s a symbol for the natural numbers."},{"Start":"01:01.759 ","End":"01:04.595","Text":"We call it N funny N,"},{"Start":"01:04.595 ","End":"01:06.920","Text":"0, 1, 2, 3, 4, etc."},{"Start":"01:06.920 ","End":"01:11.690","Text":"I\u0027ll return to the topic of 0 in a moment. Some remarks."},{"Start":"01:11.690 ","End":"01:14.240","Text":"The N stands for natural."},{"Start":"01:14.240 ","End":"01:17.000","Text":"If you don\u0027t have a funny N like this,"},{"Start":"01:17.000 ","End":"01:19.820","Text":"you could use a plain N. There was"},{"Start":"01:19.820 ","End":"01:23.480","Text":"a disagreement about whether the 0 is a natural number,"},{"Start":"01:23.480 ","End":"01:26.195","Text":"some say it is and some say it isn\u0027t."},{"Start":"01:26.195 ","End":"01:29.435","Text":"There is an ISO standard that says it is,"},{"Start":"01:29.435 ","End":"01:31.985","Text":"but not everyone abides by this standard."},{"Start":"01:31.985 ","End":"01:33.320","Text":"If I don\u0027t say otherwise,"},{"Start":"01:33.320 ","End":"01:38.540","Text":"we\u0027ll assume that 0 is in the set of natural numbers in these clips."},{"Start":"01:38.540 ","End":"01:40.940","Text":"There\u0027s actually another word, whole numbers,"},{"Start":"01:40.940 ","End":"01:44.075","Text":"say whole numbers and for sure you include the 0."},{"Start":"01:44.075 ","End":"01:46.640","Text":"On this matter you should check with your course instructor whether"},{"Start":"01:46.640 ","End":"01:49.895","Text":"the natural numbers do or don\u0027t include 0."},{"Start":"01:49.895 ","End":"01:53.390","Text":"The next special set is a set of integers."},{"Start":"01:53.390 ","End":"01:56.000","Text":"I\u0027m assuming you all know about negative numbers,"},{"Start":"01:56.000 ","End":"01:57.440","Text":"of course you will do."},{"Start":"01:57.440 ","End":"02:03.300","Text":"The integers are 0 plus 1 minus 1 plus 2 minus 2,"},{"Start":"02:03.300 ","End":"02:07.985","Text":"it\u0027s like the natural numbers except that we allow negatives also."},{"Start":"02:07.985 ","End":"02:11.570","Text":"Now the Z stands for Zahlen,"},{"Start":"02:11.570 ","End":"02:16.085","Text":"which is the German word for numbers or numerals."},{"Start":"02:16.085 ","End":"02:20.555","Text":"You can use a plane Z if you have difficulty with the funny Z."},{"Start":"02:20.555 ","End":"02:22.025","Text":"Z, if you\u0027re in England."},{"Start":"02:22.025 ","End":"02:26.950","Text":"Note that the natural numbers are a subset of the integers."},{"Start":"02:26.950 ","End":"02:29.230","Text":"Let\u0027s move on to the next set."},{"Start":"02:29.230 ","End":"02:32.545","Text":"That will be the set of rational numbers."},{"Start":"02:32.545 ","End":"02:36.230","Text":"I\u0027m assuming that you all know about fractions,"},{"Start":"02:36.230 ","End":"02:42.095","Text":"but roughly speaking, a rational number is a fraction of 2 integers, so more formally,"},{"Start":"02:42.095 ","End":"02:46.385","Text":"call it the set Q, which is the set of all p/q,"},{"Start":"02:46.385 ","End":"02:49.550","Text":"where p and q are integers,"},{"Start":"02:49.550 ","End":"02:53.200","Text":"but of course the denominator is not going to be 0."},{"Start":"02:53.200 ","End":"02:57.395","Text":"Note that a fraction can be expressed in more than 1 way,"},{"Start":"02:57.395 ","End":"03:00.845","Text":"like 1/2 is 5/10, etc."},{"Start":"03:00.845 ","End":"03:03.185","Text":"If n is an integer,"},{"Start":"03:03.185 ","End":"03:09.740","Text":"then n is also a rational number because we can always write n as n/1."},{"Start":"03:09.740 ","End":"03:13.670","Text":"Although it doesn\u0027t look like p/q it could be written as p/q."},{"Start":"03:13.670 ","End":"03:21.425","Text":"A decimal is a rational number if it is either terminating or repeating. What do I mean?"},{"Start":"03:21.425 ","End":"03:26.290","Text":"4.65289 is terminating because it comes to an end,"},{"Start":"03:26.290 ","End":"03:27.905","Text":"so that\u0027s a rational number."},{"Start":"03:27.905 ","End":"03:34.820","Text":"We can write it as this number here without the point over 100,000."},{"Start":"03:34.820 ","End":"03:41.705","Text":"Also the following is a rational number, 4.56789789789,"},{"Start":"03:41.705 ","End":"03:44.150","Text":"where the pattern is that the 789 repeats,"},{"Start":"03:44.150 ","End":"03:45.320","Text":"so because it\u0027s repeating,"},{"Start":"03:45.320 ","End":"03:47.040","Text":"it\u0027s also a rational number,"},{"Start":"03:47.040 ","End":"03:50.900","Text":"another technique of computing what the fraction is."},{"Start":"03:50.900 ","End":"03:55.305","Text":"The Q in the definition stands for quotient,"},{"Start":"03:55.305 ","End":"03:58.400","Text":"of course if you can type the funny Q,"},{"Start":"03:58.400 ","End":"04:00.335","Text":"you could just use plain Q."},{"Start":"04:00.335 ","End":"04:03.620","Text":"Note that the natural numbers are contained in"},{"Start":"04:03.620 ","End":"04:08.200","Text":"the integers and the integers are contained in the rational numbers."},{"Start":"04:08.200 ","End":"04:11.490","Text":"Next we\u0027ll talk about the real numbers,"},{"Start":"04:11.490 ","End":"04:13.430","Text":"so we\u0027ll have longer chain,"},{"Start":"04:13.430 ","End":"04:15.200","Text":"natural numbers contained in integers,"},{"Start":"04:15.200 ","End":"04:17.840","Text":"contained in rationals, contained in reals."},{"Start":"04:17.840 ","End":"04:20.135","Text":"Now we\u0027ll talk about real numbers,"},{"Start":"04:20.135 ","End":"04:22.939","Text":"and also something called the irrational numbers."},{"Start":"04:22.939 ","End":"04:27.170","Text":"Turns out that the rational numbers Q are just not enough for our purposes."},{"Start":"04:27.170 ","End":"04:28.530","Text":"I\u0027ll give you an example why."},{"Start":"04:28.530 ","End":"04:30.290","Text":"In geometry you could ask,"},{"Start":"04:30.290 ","End":"04:36.110","Text":"what\u0027s the length of the diagonal of a square with side equal to 1 as in the picture?"},{"Start":"04:36.110 ","End":"04:38.060","Text":"What\u0027s the length of this diagonal?"},{"Start":"04:38.060 ","End":"04:40.300","Text":"If you apply Pythagoras\u0027s theorem,"},{"Start":"04:40.300 ","End":"04:43.125","Text":"x satisfies x squared equals 2 squared,"},{"Start":"04:43.125 ","End":"04:45.680","Text":"which is 1 squared plus 1 squared."},{"Start":"04:45.680 ","End":"04:48.460","Text":"So we need a number whose square is 2,"},{"Start":"04:48.460 ","End":"04:53.690","Text":"but the Greeks discovered already that square root of 2 can\u0027t be a rational number."},{"Start":"04:53.690 ","End":"04:56.000","Text":"There\u0027s actually a short proof for this,"},{"Start":"04:56.000 ","End":"04:57.335","Text":"but I won\u0027t give it here,"},{"Start":"04:57.335 ","End":"05:00.130","Text":"but still it has a place on the number line,"},{"Start":"05:00.130 ","End":"05:04.370","Text":"there is a quantity that you could copy to the number line."},{"Start":"05:04.370 ","End":"05:08.030","Text":"What we do is develop the set of real numbers,"},{"Start":"05:08.030 ","End":"05:11.830","Text":"and this will correspond to the set of points on the line."},{"Start":"05:11.830 ","End":"05:15.200","Text":"When you have a line, you also need to choose a 0,"},{"Start":"05:15.200 ","End":"05:17.060","Text":"an origin and a unit of length and"},{"Start":"05:17.060 ","End":"05:20.300","Text":"a direction which is positive and the other direction is negative."},{"Start":"05:20.300 ","End":"05:21.560","Text":"We\u0027ll get into all that."},{"Start":"05:21.560 ","End":"05:24.530","Text":"This is an informal description of the set of real numbers,"},{"Start":"05:24.530 ","End":"05:26.065","Text":"the points on a line,"},{"Start":"05:26.065 ","End":"05:29.705","Text":"but it\u0027s quite difficult to give a rigorous definition,"},{"Start":"05:29.705 ","End":"05:33.470","Text":"difficult for beginners and it\u0027s not in this course anyway."},{"Start":"05:33.470 ","End":"05:36.770","Text":"Now, an irrational number is a real number,"},{"Start":"05:36.770 ","End":"05:38.755","Text":"which is not a rational number."},{"Start":"05:38.755 ","End":"05:40.110","Text":"We gave an example,"},{"Start":"05:40.110 ","End":"05:42.350","Text":"the square root of 2 is a real number,"},{"Start":"05:42.350 ","End":"05:43.610","Text":"but it\u0027s not irrational number,"},{"Start":"05:43.610 ","End":"05:44.930","Text":"and there are other famous ones,"},{"Start":"05:44.930 ","End":"05:48.210","Text":"Pi, e and Phi,"},{"Start":"05:48.210 ","End":"05:51.900","Text":"which is the golden section, the golden ratio."},{"Start":"05:51.900 ","End":"05:56.525","Text":"The set of irrational numbers is often denoted with a capital P,"},{"Start":"05:56.525 ","End":"05:58.430","Text":"but it\u0027s not universal."},{"Start":"05:58.430 ","End":"06:01.895","Text":"Now I\u0027ll give you a diagram showing the relationships of"},{"Start":"06:01.895 ","End":"06:05.540","Text":"the sets we\u0027ve studied so far, the famous ones."},{"Start":"06:05.540 ","End":"06:07.775","Text":"We have the real numbers,"},{"Start":"06:07.775 ","End":"06:12.415","Text":"and the real numbers contain rationals and irrationals."},{"Start":"06:12.415 ","End":"06:15.080","Text":"The rationals contain the integers,"},{"Start":"06:15.080 ","End":"06:18.515","Text":"and the integers contain the natural numbers."},{"Start":"06:18.515 ","End":"06:20.675","Text":"Nothing more to say here."},{"Start":"06:20.675 ","End":"06:23.075","Text":"Now let\u0027s talk about the number line."},{"Start":"06:23.075 ","End":"06:25.670","Text":"I won\u0027t say very much because I presume you\u0027re"},{"Start":"06:25.670 ","End":"06:29.030","Text":"all familiar with it, especially from geometry."},{"Start":"06:29.030 ","End":"06:31.745","Text":"Here\u0027s a picture of it."},{"Start":"06:31.745 ","End":"06:36.860","Text":"Number line is a 0 and is the positive direction indicated by the arrow,"},{"Start":"06:36.860 ","End":"06:38.280","Text":"so the other direction is negative."},{"Start":"06:38.280 ","End":"06:39.620","Text":"Then the unit of measurement,"},{"Start":"06:39.620 ","End":"06:40.940","Text":"this is 0 and this is 1,"},{"Start":"06:40.940 ","End":"06:42.890","Text":"2, 3, and so on."},{"Start":"06:42.890 ","End":"06:44.120","Text":"Here, for example,"},{"Start":"06:44.120 ","End":"06:48.155","Text":"is square root of 2, e, Pi."},{"Start":"06:48.155 ","End":"06:51.625","Text":"I\u0027d like to mention that beyond the real numbers,"},{"Start":"06:51.625 ","End":"06:53.260","Text":"we can go further."},{"Start":"06:53.260 ","End":"06:55.360","Text":"The problem with the real numbers is that there\u0027s"},{"Start":"06:55.360 ","End":"06:58.870","Text":"no solution to x squared equals minus 1."},{"Start":"06:58.870 ","End":"07:01.190","Text":"There\u0027s no square root of minus 1,"},{"Start":"07:01.190 ","End":"07:03.225","Text":"and there are complex numbers,"},{"Start":"07:03.225 ","End":"07:06.325","Text":"and the set of complex numbers is denoted with this funny C,"},{"Start":"07:06.325 ","End":"07:07.990","Text":"and this contains the real numbers,"},{"Start":"07:07.990 ","End":"07:11.035","Text":"but the equation x squared plus 1 equals 0,"},{"Start":"07:11.035 ","End":"07:13.895","Text":"or x squared equals minus 1 does have a solution,"},{"Start":"07:13.895 ","End":"07:17.525","Text":"and the solution is i,"},{"Start":"07:17.525 ","End":"07:19.840","Text":"which is a new number,"},{"Start":"07:19.840 ","End":"07:23.350","Text":"an imaginary number that belongs to C,"},{"Start":"07:23.350 ","End":"07:26.620","Text":"but that\u0027s beyond the scope of this course."},{"Start":"07:26.620 ","End":"07:28.525","Text":"I\u0027m just mentioning it."},{"Start":"07:28.525 ","End":"07:31.430","Text":"That concludes this clip."}],"ID":26556},{"Watched":false,"Name":"Exercise 5","Duration":"1m 24s","ChapterTopicVideoID":25743,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this exercise, we have to find 2 sets, A and B,"},{"Start":"00:04.980 ","End":"00:11.395","Text":"such that A is a member of B and A is a subset of B,"},{"Start":"00:11.395 ","End":"00:13.715","Text":"both of these relations hold."},{"Start":"00:13.715 ","End":"00:16.345","Text":"There are many possible examples."},{"Start":"00:16.345 ","End":"00:22.430","Text":"I\u0027ll give you a possible solution that A be the set containing 1, 2,"},{"Start":"00:22.430 ","End":"00:25.730","Text":"and B contains, first of all,"},{"Start":"00:25.730 ","End":"00:27.500","Text":"the set 1, 2,"},{"Start":"00:27.500 ","End":"00:30.770","Text":"and then the elements 1 and 2."},{"Start":"00:30.770 ","End":"00:36.275","Text":"All you have to do really is take the set A and throw it inside B,"},{"Start":"00:36.275 ","End":"00:38.510","Text":"the whole set, and then break it"},{"Start":"00:38.510 ","End":"00:42.190","Text":"open and pour its contents out and you could add some more to it,"},{"Start":"00:42.190 ","End":"00:43.984","Text":"it wouldn\u0027t make any difference."},{"Start":"00:43.984 ","End":"00:45.620","Text":"In the previous exercise,"},{"Start":"00:45.620 ","End":"00:47.840","Text":"we had an example, 2,"},{"Start":"00:47.840 ","End":"00:54.960","Text":"4 is a subset of this set A and it\u0027s also a member because 2,"},{"Start":"00:54.960 ","End":"00:56.190","Text":"4 is the same as 4,"},{"Start":"00:56.190 ","End":"00:58.745","Text":"2, so that\u0027s an example."},{"Start":"00:58.745 ","End":"01:01.370","Text":"Now I\u0027ll give you 1 more example."},{"Start":"01:01.370 ","End":"01:07.090","Text":"Take A as the empty set and B as the set containing the empty set,"},{"Start":"01:07.090 ","End":"01:09.950","Text":"so A is a member of B,"},{"Start":"01:09.950 ","End":"01:16.685","Text":"because here is A and A is a subset of B because the empty set is a subset of every set,"},{"Start":"01:16.685 ","End":"01:18.600","Text":"everything in A is also in B,"},{"Start":"01:18.600 ","End":"01:24.540","Text":"and that\u0027s vacuously true because there is nothing in A. We\u0027re done."}],"ID":26547},{"Watched":false,"Name":"Exercise 6","Duration":"3m 43s","ChapterTopicVideoID":25744,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.830","Text":"In this exercise,"},{"Start":"00:01.830 ","End":"00:03.855","Text":"we are given 5 sets,"},{"Start":"00:03.855 ","End":"00:06.735","Text":"A, B, C, D, E as follows,"},{"Start":"00:06.735 ","End":"00:10.185","Text":"and is a 3 part question,"},{"Start":"00:10.185 ","End":"00:15.510","Text":"which of these sets could be X in each of these?"},{"Start":"00:15.510 ","End":"00:18.870","Text":"We\u0027ll read each one as we solve it."},{"Start":"00:18.870 ","End":"00:20.880","Text":"In part a,"},{"Start":"00:20.880 ","End":"00:24.180","Text":"we\u0027re given that X is a subset of A,"},{"Start":"00:24.180 ","End":"00:26.985","Text":"but X is not a subset of D,"},{"Start":"00:26.985 ","End":"00:30.240","Text":"and X is one of these."},{"Start":"00:30.240 ","End":"00:34.380","Text":"What I suggest, is we just go over them one by one and"},{"Start":"00:34.380 ","End":"00:39.186","Text":"see which one is a subset of A but not a subset of D."},{"Start":"00:39.186 ","End":"00:42.086","Text":"Now A is certainly a subset of A"},{"Start":"00:42.086 ","End":"00:45.510","Text":"because every set is a subset of itself and it\u0027s"},{"Start":"00:45.510 ","End":"00:51.255","Text":"also not a subset of D. This is one possibility, I\u0027ll mark it."},{"Start":"00:51.255 ","End":"00:52.935","Text":"Now, the next one,"},{"Start":"00:52.935 ","End":"00:55.635","Text":"is this a subset of A?"},{"Start":"00:55.635 ","End":"00:59.490","Text":"No, because 10 is in here,"},{"Start":"00:59.490 ","End":"01:01.920","Text":"but not in here, so that\u0027s not a subset of A,"},{"Start":"01:01.920 ","End":"01:04.725","Text":"so don\u0027t have to look at the second part."},{"Start":"01:04.725 ","End":"01:08.490","Text":"C, is it a subset of A?"},{"Start":"01:08.490 ","End":"01:11.070","Text":"Yeah, everything in here is in here."},{"Start":"01:11.070 ","End":"01:13.500","Text":"Is it a subset of D?"},{"Start":"01:13.500 ","End":"01:19.545","Text":"Well, no, 9 is in C but not in D for example. That\u0027s good."},{"Start":"01:19.545 ","End":"01:22.530","Text":"It\u0027s a subset of A but not of D,"},{"Start":"01:22.530 ","End":"01:25.590","Text":"so that\u0027s another possibility."},{"Start":"01:25.590 ","End":"01:30.255","Text":"Now, D, it\u0027s a subset of A,"},{"Start":"01:30.255 ","End":"01:34.170","Text":"but it is a subset of D, so no good."},{"Start":"01:34.170 ","End":"01:37.575","Text":"E is a subset of A, yes,"},{"Start":"01:37.575 ","End":"01:39.480","Text":"and it is a subset of D,"},{"Start":"01:39.480 ","End":"01:41.160","Text":"so also no good."},{"Start":"01:41.160 ","End":"01:46.470","Text":"The only possibilities are A or C and that\u0027s part a."},{"Start":"01:46.470 ","End":"01:48.870","Text":"Let\u0027s move on to part b."},{"Start":"01:48.870 ","End":"01:54.080","Text":"X is a subset of D and X is not a subset of C."},{"Start":"01:54.080 ","End":"01:59.160","Text":"Once again, we\u0027ll go over them one by one,"},{"Start":"01:59.160 ","End":"02:02.259","Text":"over A, B, C, D, E."},{"Start":"02:02.259 ","End":"02:06.570","Text":"Start with A is a subset of D?"},{"Start":"02:06.570 ","End":"02:09.570","Text":"Nope. Let\u0027s try the next one."},{"Start":"02:09.570 ","End":"02:11.970","Text":"B, is it a subset of D?"},{"Start":"02:11.970 ","End":"02:14.880","Text":"No. Let\u0027s try C,"},{"Start":"02:14.880 ","End":"02:16.365","Text":"is it a subset of D?"},{"Start":"02:16.365 ","End":"02:20.280","Text":"No. D, is it a subset of D?"},{"Start":"02:20.280 ","End":"02:23.865","Text":"Yes. Is it a subset of C?"},{"Start":"02:23.865 ","End":"02:25.740","Text":"The answer is no."},{"Start":"02:25.740 ","End":"02:27.720","Text":"Because for example, 6 is in here,"},{"Start":"02:27.720 ","End":"02:30.375","Text":"not in here, so this looks good."},{"Start":"02:30.375 ","End":"02:32.085","Text":"We\u0027ll mark this."},{"Start":"02:32.085 ","End":"02:33.540","Text":"Now let\u0027s try E,"},{"Start":"02:33.540 ","End":"02:35.790","Text":"is it a subset of D?"},{"Start":"02:35.790 ","End":"02:36.945","Text":"Yes, it is."},{"Start":"02:36.945 ","End":"02:39.090","Text":"Both 7 and 8 are in here."},{"Start":"02:39.090 ","End":"02:41.295","Text":"Is E a subset of C,"},{"Start":"02:41.295 ","End":"02:44.115","Text":"no, 8 is in here and is not in here."},{"Start":"02:44.115 ","End":"02:46.245","Text":"This is also good."},{"Start":"02:46.245 ","End":"02:49.650","Text":"The 2 answers are D or E,"},{"Start":"02:49.650 ","End":"02:52.140","Text":"and that\u0027s part b."},{"Start":"02:52.140 ","End":"02:54.840","Text":"Now on to part c,"},{"Start":"02:54.840 ","End":"02:57.780","Text":"we want X to be a subset of E,"},{"Start":"02:57.780 ","End":"03:00.375","Text":"but not a subset of A."},{"Start":"03:00.375 ","End":"03:02.925","Text":"It\u0027s got to be one of these."},{"Start":"03:02.925 ","End":"03:05.250","Text":"We\u0027ll just try them one by one."},{"Start":"03:05.250 ","End":"03:08.055","Text":"Let\u0027s start with A."},{"Start":"03:08.055 ","End":"03:10.740","Text":"Is it a subset of E?"},{"Start":"03:10.740 ","End":"03:14.040","Text":"No. We can continue."},{"Start":"03:14.040 ","End":"03:16.500","Text":"B, is it a subset of E?"},{"Start":"03:16.500 ","End":"03:19.785","Text":"Nope. Here a subset of E?"},{"Start":"03:19.785 ","End":"03:22.785","Text":"Nope. This is a subset of E?"},{"Start":"03:22.785 ","End":"03:25.530","Text":"Nope. E, subset of E?"},{"Start":"03:25.530 ","End":"03:29.580","Text":"Yes. Is it a subset of A?"},{"Start":"03:29.580 ","End":"03:33.240","Text":"Yes, 7 and 8 are both here."},{"Start":"03:33.240 ","End":"03:35.385","Text":"It\u0027s not good."},{"Start":"03:35.385 ","End":"03:37.695","Text":"We didn\u0027t find anything."},{"Start":"03:37.695 ","End":"03:40.170","Text":"X can\u0027t be any of them."},{"Start":"03:40.170 ","End":"03:43.360","Text":"That\u0027s part c and we\u0027re done."}],"ID":26548},{"Watched":false,"Name":"Exercise 7","Duration":"1m 49s","ChapterTopicVideoID":25745,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this exercise, we have to prove that a subset"},{"Start":"00:03.600 ","End":"00:06.930","Text":"of a subset is a subset. Let me spell it out."},{"Start":"00:06.930 ","End":"00:11.550","Text":"If A is a subset of B and B is a subset of C,"},{"Start":"00:11.550 ","End":"00:19.665","Text":"then A is a subset of C. We also say that the subset relation is transitive."},{"Start":"00:19.665 ","End":"00:22.180","Text":"Think of it like less than or equal to,"},{"Start":"00:22.180 ","End":"00:24.120","Text":"in algebra if A is less than or equal to B,"},{"Start":"00:24.120 ","End":"00:25.470","Text":"and B is less than or equal to C,"},{"Start":"00:25.470 ","End":"00:28.725","Text":"then A is less than or equal to C. That\u0027s just a mnemonic."},{"Start":"00:28.725 ","End":"00:31.620","Text":"Let\u0027s get to proving it."},{"Start":"00:31.620 ","End":"00:35.195","Text":"We\u0027re given this and this,"},{"Start":"00:35.195 ","End":"00:37.910","Text":"and we have to show this,"},{"Start":"00:37.910 ","End":"00:41.360","Text":"now we need to remember the definition of subset ."},{"Start":"00:41.360 ","End":"00:45.110","Text":"In general, we say that X is a subset of Y if and"},{"Start":"00:45.110 ","End":"00:48.810","Text":"only if this is equivalent to an implication,"},{"Start":"00:48.810 ","End":"00:53.620","Text":"and if then that if an element x belongs to X,"},{"Start":"00:53.620 ","End":"00:56.405","Text":"then the same element belongs to Y."},{"Start":"00:56.405 ","End":"00:58.085","Text":"This is what we have to show,"},{"Start":"00:58.085 ","End":"01:03.229","Text":"and this is equivalent to showing that x belongs to A implies"},{"Start":"01:03.229 ","End":"01:08.690","Text":"x belongs to C . We\u0027ll start from this and we\u0027ll derive this."},{"Start":"01:08.690 ","End":"01:10.325","Text":"This is where we start."},{"Start":"01:10.325 ","End":"01:14.735","Text":"I claim that x belongs to B."},{"Start":"01:14.735 ","End":"01:17.960","Text":"This is because A is a subset of B."},{"Start":"01:17.960 ","End":"01:22.795","Text":"If you take this definition here with A and B,"},{"Start":"01:22.795 ","End":"01:27.335","Text":"then we get that x belongs to B."},{"Start":"01:27.335 ","End":"01:29.540","Text":"It belongs to X,"},{"Start":"01:29.540 ","End":"01:31.190","Text":"which is A, therefore belongs to Y,"},{"Start":"01:31.190 ","End":"01:32.585","Text":"which is B in our case."},{"Start":"01:32.585 ","End":"01:36.650","Text":"Now from this, I can deduce that x belongs to C. Once again,"},{"Start":"01:36.650 ","End":"01:38.585","Text":"we use this template,"},{"Start":"01:38.585 ","End":"01:42.210","Text":"but this time with B and C, X belongs to B,"},{"Start":"01:42.210 ","End":"01:45.350","Text":"therefore x belongs to C because B is"},{"Start":"01:45.350 ","End":"01:50.460","Text":"a subset of C. This is what we had to show. We\u0027re done."}],"ID":26549},{"Watched":false,"Name":"Intervals on the Number Line","Duration":"3m 59s","ChapterTopicVideoID":25749,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.245","Text":"Now, let\u0027s talk about intervals on the number line."},{"Start":"00:04.245 ","End":"00:07.275","Text":"An interval is a special set of real numbers."},{"Start":"00:07.275 ","End":"00:10.785","Text":"It\u0027s very important and used a lot in calculus."},{"Start":"00:10.785 ","End":"00:12.285","Text":"You can\u0027t do without it."},{"Start":"00:12.285 ","End":"00:14.700","Text":"I\u0027ll start with an example."},{"Start":"00:14.700 ","End":"00:16.920","Text":"Let\u0027s take a look at the picture first."},{"Start":"00:16.920 ","End":"00:22.590","Text":"Suppose I want all the numbers from 1-4,"},{"Start":"00:22.590 ","End":"00:23.940","Text":"including the 1,"},{"Start":"00:23.940 ","End":"00:25.755","Text":"but not including the 4."},{"Start":"00:25.755 ","End":"00:31.230","Text":"Pictorially, we put a solid dot here to show that the 1 is included,"},{"Start":"00:31.230 ","End":"00:35.370","Text":"but a hollow dot here to say that the 4 isn\u0027t included."},{"Start":"00:35.370 ","End":"00:38.100","Text":"We could describe it by an inequality to"},{"Start":"00:38.100 ","End":"00:41.235","Text":"say what x has to satisfy to be on this interval."},{"Start":"00:41.235 ","End":"00:44.520","Text":"But clearly, it\u0027s x bigger or equal to 1,"},{"Start":"00:44.520 ","End":"00:46.605","Text":"but strictly, less than 4."},{"Start":"00:46.605 ","End":"00:53.630","Text":"We write this as 1,4 with a square bracket on the left to say that the 1 is included,"},{"Start":"00:53.630 ","End":"00:58.670","Text":"and the round bracket by the 4 to say that the 4 is not included."},{"Start":"00:58.670 ","End":"01:01.160","Text":"This is what we call a half-open interval."},{"Start":"01:01.160 ","End":"01:04.010","Text":"The open refers to the round bracket,"},{"Start":"01:04.010 ","End":"01:07.450","Text":"and the close refers to the square bracket."},{"Start":"01:07.450 ","End":"01:09.860","Text":"This interval has the inequality,"},{"Start":"01:09.860 ","End":"01:12.395","Text":"and we also have a set notation."},{"Start":"01:12.395 ","End":"01:15.650","Text":"We described the interval of the set of all x such"},{"Start":"01:15.650 ","End":"01:19.045","Text":"that 1 less than or equal to x less than 4."},{"Start":"01:19.045 ","End":"01:22.790","Text":"Now, another example, this time, an infinite interval."},{"Start":"01:22.790 ","End":"01:26.840","Text":"The interval from minus infinity to 3."},{"Start":"01:26.840 ","End":"01:29.165","Text":"The minus infinity, it\u0027s just symbolic."},{"Start":"01:29.165 ","End":"01:31.265","Text":"It means unbounded below."},{"Start":"01:31.265 ","End":"01:33.905","Text":"We imagine it is down to minus infinity,"},{"Start":"01:33.905 ","End":"01:38.120","Text":"so it has the inequality x less than 3, or often,"},{"Start":"01:38.120 ","End":"01:43.555","Text":"we include the infinity by saying minus infinity less than x, less than 3."},{"Start":"01:43.555 ","End":"01:46.250","Text":"The set notation instead of all x,"},{"Start":"01:46.250 ","End":"01:48.440","Text":"sometimes we emphasize that it\u0027s in"},{"Start":"01:48.440 ","End":"01:51.410","Text":"the reals by writing belongs to R. Usually, it\u0027s omitted."},{"Start":"01:51.410 ","End":"01:53.870","Text":"Set of all x such that x is less than 3,"},{"Start":"01:53.870 ","End":"01:58.760","Text":"or the set of all x such that x is between minus infinity and 3."},{"Start":"01:58.760 ","End":"02:02.405","Text":"This is sometimes called a semi-infinite interval."},{"Start":"02:02.405 ","End":"02:04.070","Text":"It\u0027s an open interval by the way."},{"Start":"02:04.070 ","End":"02:08.155","Text":"There\u0027s an actual infinite interval is the whole number line."},{"Start":"02:08.155 ","End":"02:11.150","Text":"We write that as minus infinity to"},{"Start":"02:11.150 ","End":"02:16.040","Text":"infinity or the set of all x such that x is between minus infinity to infinity,"},{"Start":"02:16.040 ","End":"02:18.245","Text":"which really means all x."},{"Start":"02:18.245 ","End":"02:20.465","Text":"Well, no condition."},{"Start":"02:20.465 ","End":"02:24.455","Text":"Now, let\u0027s go through all the different types of intervals."},{"Start":"02:24.455 ","End":"02:26.195","Text":"Just do this quickly."},{"Start":"02:26.195 ","End":"02:27.740","Text":"For each type of interval,"},{"Start":"02:27.740 ","End":"02:31.940","Text":"we have an inequality interval notation,"},{"Start":"02:31.940 ","End":"02:34.085","Text":"a bit of the graph and a description."},{"Start":"02:34.085 ","End":"02:35.990","Text":"Let\u0027s take this one."},{"Start":"02:35.990 ","End":"02:38.690","Text":"This one is finite and open."},{"Start":"02:38.690 ","End":"02:40.580","Text":"It\u0027s open on both sides."},{"Start":"02:40.580 ","End":"02:41.690","Text":"Open, as I said,"},{"Start":"02:41.690 ","End":"02:45.355","Text":"means not includes the end round brackets."},{"Start":"02:45.355 ","End":"02:49.250","Text":"We could describe this as x is strictly between a and b."},{"Start":"02:49.250 ","End":"02:50.855","Text":"This is the inequality,"},{"Start":"02:50.855 ","End":"02:55.865","Text":"and this is how the graph looks like on the number line, so one-dimensional graph."},{"Start":"02:55.865 ","End":"02:58.630","Text":"Then we have the half-open interval."},{"Start":"02:58.630 ","End":"03:02.135","Text":"It\u0027s half open on the right to be strict."},{"Start":"03:02.135 ","End":"03:03.500","Text":"On the right, it\u0027s open."},{"Start":"03:03.500 ","End":"03:05.290","Text":"On the left, it\u0027s closed."},{"Start":"03:05.290 ","End":"03:07.580","Text":"This is what it looks like, and we have the reverse,"},{"Start":"03:07.580 ","End":"03:11.060","Text":"the half-open interval, where x is between a and b."},{"Start":"03:11.060 ","End":"03:14.210","Text":"Includes b, so it\u0027s half open on the left,"},{"Start":"03:14.210 ","End":"03:16.585","Text":"or open on the left, closed on the right."},{"Start":"03:16.585 ","End":"03:20.224","Text":"Here we have the closed interval, finite and closed."},{"Start":"03:20.224 ","End":"03:21.830","Text":"It\u0027s from a to b,"},{"Start":"03:21.830 ","End":"03:23.870","Text":"including a and b."},{"Start":"03:23.870 ","End":"03:27.670","Text":"Then we have the infinite intervals."},{"Start":"03:27.670 ","End":"03:30.320","Text":"We have this one,"},{"Start":"03:30.320 ","End":"03:32.750","Text":"a round bracket infinity,"},{"Start":"03:32.750 ","End":"03:34.760","Text":"from a to infinity,"},{"Start":"03:34.760 ","End":"03:38.460","Text":"but really, it\u0027s x greater than a,"},{"Start":"03:38.810 ","End":"03:42.785","Text":"though we sometimes write less than infinity."},{"Start":"03:42.785 ","End":"03:44.755","Text":"This is the graph."},{"Start":"03:44.755 ","End":"03:48.165","Text":"Then x less than a, like so."},{"Start":"03:48.165 ","End":"03:49.770","Text":"X bigger or equal to a,"},{"Start":"03:49.770 ","End":"03:51.050","Text":"so we have a square bracket,"},{"Start":"03:51.050 ","End":"03:55.220","Text":"includes the a and x less than or equal to a, like so."},{"Start":"03:55.220 ","End":"03:57.170","Text":"I\u0027ll leave you to look at this,"},{"Start":"03:57.170 ","End":"04:00.450","Text":"and we are done for this clip."}],"ID":26553},{"Watched":false,"Name":"Exercise 8","Duration":"3m 22s","ChapterTopicVideoID":25746,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.495","Text":"In this exercise, we have 6 parts."},{"Start":"00:03.495 ","End":"00:06.690","Text":"We have to rewrite each of the set descriptions either as"},{"Start":"00:06.690 ","End":"00:11.400","Text":"an interval or by listing its elements in curly braces, whichever is appropriate."},{"Start":"00:11.400 ","End":"00:14.475","Text":"We also have to find the cardinality of the set,"},{"Start":"00:14.475 ","End":"00:18.160","Text":"and read each 1 and solve it as we come to it."},{"Start":"00:18.160 ","End":"00:24.660","Text":"Part A is the set of all x such that x squared is less than 16."},{"Start":"00:24.660 ","End":"00:28.440","Text":"Well, think about it, x squared less than 16,"},{"Start":"00:28.440 ","End":"00:30.840","Text":"you think of the right way up parabola,"},{"Start":"00:30.840 ","End":"00:33.900","Text":"it hits the x axis at 4 and minus 4,"},{"Start":"00:33.900 ","End":"00:37.705","Text":"and below the axis between minus 4 and 4."},{"Start":"00:37.705 ","End":"00:39.740","Text":"This is the interval,"},{"Start":"00:39.740 ","End":"00:42.895","Text":"and we can also write it as minus 4, 4."},{"Start":"00:42.895 ","End":"00:45.315","Text":"The cardinality of the set is infinite,"},{"Start":"00:45.315 ","End":"00:47.675","Text":"infinite set of points in an interval."},{"Start":"00:47.675 ","End":"00:51.455","Text":"The next 1 is that of all whole numbers x,"},{"Start":"00:51.455 ","End":"00:53.810","Text":"such that x squared is less than 16."},{"Start":"00:53.810 ","End":"00:55.870","Text":"Here, it\u0027s different."},{"Start":"00:55.870 ","End":"01:00.800","Text":"Here, we also want x to be between minus 4 and 4,"},{"Start":"01:00.800 ","End":"01:06.690","Text":"but x is an integer so we just get the following list: minus 3,"},{"Start":"01:06.690 ","End":"01:08.490","Text":"minus 2, minus 1, 0, 1, 2,"},{"Start":"01:08.490 ","End":"01:10.040","Text":"3, and of course,"},{"Start":"01:10.040 ","End":"01:12.695","Text":"there are 7 elements here."},{"Start":"01:12.695 ","End":"01:18.545","Text":"Now part C, x is a natural number such that x squared is less than 16."},{"Start":"01:18.545 ","End":"01:22.120","Text":"This time we\u0027ll get even less numbers."},{"Start":"01:22.120 ","End":"01:25.870","Text":"We still have the inequality between minus 4 and 4,"},{"Start":"01:25.870 ","End":"01:32.270","Text":"but there\u0027s only 4 natural numbers between minus 4 and 4 not including the end points,"},{"Start":"01:32.270 ","End":"01:34.925","Text":"and the cardinality is 4."},{"Start":"01:34.925 ","End":"01:41.340","Text":"However, I mentioned that some people don\u0027t include 0 as a natural number,"},{"Start":"01:41.340 ","End":"01:43.134","Text":"and if that\u0027s the case,"},{"Start":"01:43.134 ","End":"01:44.980","Text":"then the answer will come out differently."},{"Start":"01:44.980 ","End":"01:49.585","Text":"We won\u0027t include the 0 here and the answer will be 3 here."},{"Start":"01:49.585 ","End":"01:51.610","Text":"Now Part D,"},{"Start":"01:51.610 ","End":"01:53.620","Text":"we want the set of all x,"},{"Start":"01:53.620 ","End":"01:56.350","Text":"which is an integer such that x plus 4,"},{"Start":"01:56.350 ","End":"01:59.080","Text":"x minus 1 is less than 0."},{"Start":"01:59.080 ","End":"02:02.495","Text":"This comes out using the parabolas,"},{"Start":"02:02.495 ","End":"02:05.870","Text":"it hits at minus 4 and 1 and we want below the graph,"},{"Start":"02:05.870 ","End":"02:08.495","Text":"so it\u0027s between minus 4 and 1,"},{"Start":"02:08.495 ","End":"02:10.755","Text":"but it\u0027s integers,"},{"Start":"02:10.755 ","End":"02:13.425","Text":"so we just have the following integers,"},{"Start":"02:13.425 ","End":"02:16.500","Text":"and the cardinality is 4."},{"Start":"02:16.500 ","End":"02:24.410","Text":"Part E, natural numbers x such that x cubed plus x squared minus 2x is 0."},{"Start":"02:24.410 ","End":"02:28.175","Text":"We can factorize, take the x out."},{"Start":"02:28.175 ","End":"02:31.010","Text":"This factorizes further into x minus 1,"},{"Start":"02:31.010 ","End":"02:32.330","Text":"x plus 2,"},{"Start":"02:32.330 ","End":"02:35.190","Text":"so x will be the 0,"},{"Start":"02:35.190 ","End":"02:37.000","Text":"1 or minus 2."},{"Start":"02:37.000 ","End":"02:42.910","Text":"But the minus 2 won\u0027t be included because we\u0027re talking about natural numbers,"},{"Start":"02:42.910 ","End":"02:45.670","Text":"and the cardinality of this is 2."},{"Start":"02:45.670 ","End":"02:49.970","Text":"Now just in case you\u0027re not including 0 as a natural number,"},{"Start":"02:49.970 ","End":"02:52.250","Text":"then this 0 won\u0027t be included here."},{"Start":"02:52.250 ","End":"02:56.350","Text":"We\u0027ll just get the set 1 and the cardinality will be 1."},{"Start":"02:56.350 ","End":"03:03.890","Text":"Last 1, x is an integer and x squared is less than or equal to 16."},{"Start":"03:03.890 ","End":"03:05.975","Text":"Once again, we have the same parabola,"},{"Start":"03:05.975 ","End":"03:08.000","Text":"x is going to be between minus 4 and 4,"},{"Start":"03:08.000 ","End":"03:12.240","Text":"but this time it includes the minus 4 and 4 and it\u0027s an integer."},{"Start":"03:12.240 ","End":"03:15.290","Text":"These are the possibilities from minus 4 to 4,"},{"Start":"03:15.290 ","End":"03:16.790","Text":"and if you count them,"},{"Start":"03:16.790 ","End":"03:19.205","Text":"the twice 4 plus 1 is 9 of them,"},{"Start":"03:19.205 ","End":"03:22.980","Text":"and that\u0027s the last 1 and we\u0027re done."}],"ID":26550},{"Watched":false,"Name":"Exercise 9","Duration":"1m 53s","ChapterTopicVideoID":25747,"CourseChapterTopicPlaylistID":198,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"This exercise, we have to write each of the following set descriptions,"},{"Start":"00:04.440 ","End":"00:09.510","Text":"either as a list in curly braces or in the form the set of all x belonging to"},{"Start":"00:09.510 ","End":"00:14.730","Text":"some set S such x satisfies some property P. See what the first one is."},{"Start":"00:14.730 ","End":"00:16.650","Text":"Set of positive odd whole numbers."},{"Start":"00:16.650 ","End":"00:19.845","Text":"Here, whole numbers means natural numbers,"},{"Start":"00:19.845 ","End":"00:23.370","Text":"and we\u0027ll assume that 0 is a natural number."},{"Start":"00:23.370 ","End":"00:28.800","Text":"Part a, we can write each odd number as 2n plus 1,"},{"Start":"00:28.800 ","End":"00:30.540","Text":"or n is some natural number."},{"Start":"00:30.540 ","End":"00:33.720","Text":"If you think about it, when n is 0, we get 1."},{"Start":"00:33.720 ","End":"00:36.030","Text":"When n is 1, we get 3 and 5,"},{"Start":"00:36.030 ","End":"00:37.680","Text":"7, and so on."},{"Start":"00:37.680 ","End":"00:39.695","Text":"This is how we could write it."},{"Start":"00:39.695 ","End":"00:44.105","Text":"But if you assume that 0 is not a natural number,"},{"Start":"00:44.105 ","End":"00:47.640","Text":"then you can write it as 2n minus 1 instead,"},{"Start":"00:47.640 ","End":"00:49.500","Text":"because n starts at 1."},{"Start":"00:49.500 ","End":"00:51.810","Text":"So twice 1 minus 1 gives us 1,"},{"Start":"00:51.810 ","End":"00:55.425","Text":"and then twice 2 minus 1 gives us 3, and so on."},{"Start":"00:55.425 ","End":"00:59.795","Text":"The next one, you want the set of prime numbers between 10 and 20,"},{"Start":"00:59.795 ","End":"01:03.900","Text":"and we could just write that as a list of 4 of them: 11,"},{"Start":"01:03.900 ","End":"01:06.360","Text":"13, 17, and 19."},{"Start":"01:06.360 ","End":"01:08.580","Text":"Next, we have points in a plane."},{"Start":"01:08.580 ","End":"01:13.795","Text":"The set of points which lie on the circle of radius 4 centered at the origin,"},{"Start":"01:13.795 ","End":"01:16.775","Text":"assuming you know some basic analytic geometry."},{"Start":"01:16.775 ","End":"01:19.550","Text":"We could write that as the set of all x, y,"},{"Start":"01:19.550 ","End":"01:23.815","Text":"pairs of real numbers belong to R squared,"},{"Start":"01:23.815 ","End":"01:28.520","Text":"such that x squared plus y squared equals 4 squared,"},{"Start":"01:28.520 ","End":"01:30.185","Text":"or we write 16 here."},{"Start":"01:30.185 ","End":"01:32.030","Text":"If you don\u0027t know about R squared,"},{"Start":"01:32.030 ","End":"01:35.630","Text":"set of all x and y such that x squared plus y squared equals 4 squared,"},{"Start":"01:35.630 ","End":"01:37.840","Text":"and x is real, and y is real."},{"Start":"01:37.840 ","End":"01:40.820","Text":"The last one, set of squares of the numbers 1,"},{"Start":"01:40.820 ","End":"01:42.670","Text":"2, 3, and 4."},{"Start":"01:42.670 ","End":"01:45.780","Text":"That comes out as 1 squared,"},{"Start":"01:45.780 ","End":"01:48.665","Text":"2 squared, 3 squared, and 4 squared."},{"Start":"01:48.665 ","End":"01:51.695","Text":"Just write it as a list in curly braces."},{"Start":"01:51.695 ","End":"01:54.360","Text":"That\u0027s it. We\u0027re done."}],"ID":26551}],"Thumbnail":null,"ID":198},{"Name":"Operations on Sets","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Union and Intersection","Duration":"7m 33s","ChapterTopicVideoID":25761,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"Next we learn about some operations that we can perform on sets,"},{"Start":"00:04.515 ","End":"00:06.420","Text":"and we\u0027ll start with the first example,"},{"Start":"00:06.420 ","End":"00:09.690","Text":"which will be the union and it\u0027s a binary operation,"},{"Start":"00:09.690 ","End":"00:12.555","Text":"it takes 2 sets and produces another set."},{"Start":"00:12.555 ","End":"00:14.310","Text":"I\u0027ll give an example."},{"Start":"00:14.310 ","End":"00:16.590","Text":"Suppose A is the set 1, 2, 5,"},{"Start":"00:16.590 ","End":"00:20.265","Text":"10, and B is the set 1, 2, 3, 4."},{"Start":"00:20.265 ","End":"00:23.850","Text":"Let\u0027s define a new set which contains all the elements that"},{"Start":"00:23.850 ","End":"00:27.180","Text":"are in A or in B. I\u0027m writing it as and,"},{"Start":"00:27.180 ","End":"00:29.320","Text":"or because or is a bit ambiguous,"},{"Start":"00:29.320 ","End":"00:31.650","Text":"so I mean 1 or the other or both."},{"Start":"00:31.650 ","End":"00:37.335","Text":"What we\u0027ll get will be the union of A and B and will be denoted A union B,"},{"Start":"00:37.335 ","End":"00:39.385","Text":"and let\u0027s see what it is,"},{"Start":"00:39.385 ","End":"00:41.015","Text":"1 is in here,"},{"Start":"00:41.015 ","End":"00:42.455","Text":"2 is in here,"},{"Start":"00:42.455 ","End":"00:44.525","Text":"also 5 and 10,"},{"Start":"00:44.525 ","End":"00:45.830","Text":"but looking at B,"},{"Start":"00:45.830 ","End":"00:47.210","Text":"1 and 2 we\u0027ve already taken,"},{"Start":"00:47.210 ","End":"00:48.730","Text":"but also 3 and 4."},{"Start":"00:48.730 ","End":"00:50.075","Text":"If you think about it,"},{"Start":"00:50.075 ","End":"00:53.660","Text":"all the elements that are either in here or in here or in both are 1,"},{"Start":"00:53.660 ","End":"00:55.475","Text":"2, 3, 4, 5 and 10."},{"Start":"00:55.475 ","End":"00:58.640","Text":"I can show you a more systematic way of doing it if you like."},{"Start":"00:58.640 ","End":"00:59.930","Text":"We could take 1, 2,"},{"Start":"00:59.930 ","End":"01:01.730","Text":"5, 10, union 1, 2, 3,"},{"Start":"01:01.730 ","End":"01:05.145","Text":"4 and say all the 1s that are in here,"},{"Start":"01:05.145 ","End":"01:09.725","Text":"then combine them with all the 1s that are in here and just write them like so."},{"Start":"01:09.725 ","End":"01:13.190","Text":"Now, there are duplicates which you want to eliminate, don\u0027t have to."},{"Start":"01:13.190 ","End":"01:14.750","Text":"This would be a correct answer,"},{"Start":"01:14.750 ","End":"01:19.345","Text":"but usually we write them without duplicates and often we order them."},{"Start":"01:19.345 ","End":"01:21.555","Text":"We have the 1 and the 2,"},{"Start":"01:21.555 ","End":"01:22.950","Text":"and that\u0027s actually in both."},{"Start":"01:22.950 ","End":"01:24.210","Text":"so I colored them this way,"},{"Start":"01:24.210 ","End":"01:27.200","Text":"then we have 3 and 4 that are just in B,"},{"Start":"01:27.200 ","End":"01:31.175","Text":"and then we have 5 and 10 that are just in A, so same answer."},{"Start":"01:31.175 ","End":"01:40.010","Text":"Formally, we define the union as set of all x such that x is in A or x is in B,"},{"Start":"01:40.010 ","End":"01:45.835","Text":"which means that x belongs to A union B if and only if x is in A or x is in B,"},{"Start":"01:45.835 ","End":"01:49.110","Text":"and the or is a non-exclusive or,"},{"Start":"01:49.110 ","End":"01:51.000","Text":"like I said, the and, or."},{"Start":"01:51.000 ","End":"01:53.195","Text":"Or includes possibility of both."},{"Start":"01:53.195 ","End":"01:54.965","Text":"Now some more examples."},{"Start":"01:54.965 ","End":"01:58.160","Text":"If we take any set, let\u0027s say 1, 2, 3,"},{"Start":"01:58.160 ","End":"02:02.630","Text":"union with itself set of all elements that are in here or in here is just the 1, 2, 3."},{"Start":"02:02.630 ","End":"02:05.330","Text":"We don\u0027t get anything extra by combining 2 of them."},{"Start":"02:05.330 ","End":"02:07.205","Text":"That would be true for any set."},{"Start":"02:07.205 ","End":"02:09.170","Text":"Union with itself is itself."},{"Start":"02:09.170 ","End":"02:11.345","Text":"Another example, 1, 4, 10."},{"Start":"02:11.345 ","End":"02:12.740","Text":"Union 1, 4, 11,"},{"Start":"02:12.740 ","End":"02:17.850","Text":"the 1 and 4 are the common elements and 10 from here and 11 from here."},{"Start":"02:17.960 ","End":"02:21.780","Text":"Now something different, 2 intervals."},{"Start":"02:21.780 ","End":"02:23.925","Text":"The interval from 1-5,"},{"Start":"02:23.925 ","End":"02:25.965","Text":"that\u0027s closed union,"},{"Start":"02:25.965 ","End":"02:28.215","Text":"the open interval from 4-7,"},{"Start":"02:28.215 ","End":"02:30.060","Text":"we need a picture for this,"},{"Start":"02:30.060 ","End":"02:35.300","Text":"from 1-5 close, that\u0027s this interval,"},{"Start":"02:35.300 ","End":"02:36.830","Text":"including the 1 in the 5,"},{"Start":"02:36.830 ","End":"02:38.830","Text":"from 4-7 open,"},{"Start":"02:38.830 ","End":"02:40.990","Text":"it doesn\u0027t include the 4 and the 7."},{"Start":"02:40.990 ","End":"02:43.700","Text":"But if we look at anything that\u0027s either in here or in here,"},{"Start":"02:43.700 ","End":"02:45.845","Text":"it\u0027s like superposing these 2."},{"Start":"02:45.845 ","End":"02:50.900","Text":"We get everything from 1 including the 1 up to 7, not including,"},{"Start":"02:50.900 ","End":"02:52.505","Text":"so we get 1,"},{"Start":"02:52.505 ","End":"02:57.695","Text":"7 half open and I\u0027ll list some properties of the union."},{"Start":"02:57.695 ","End":"03:01.220","Text":"Go over these quickly because they are fairly straightforward."},{"Start":"03:01.220 ","End":"03:04.340","Text":"The union of A and B is the union of B and A."},{"Start":"03:04.340 ","End":"03:06.020","Text":"A union B and B union A."},{"Start":"03:06.020 ","End":"03:07.880","Text":"We also have associativity,"},{"Start":"03:07.880 ","End":"03:09.140","Text":"we take A union B,"},{"Start":"03:09.140 ","End":"03:11.795","Text":"and then union C is the same as A union."},{"Start":"03:11.795 ","End":"03:13.895","Text":"The result of B union C,"},{"Start":"03:13.895 ","End":"03:15.520","Text":"like we saw here,"},{"Start":"03:15.520 ","End":"03:20.150","Text":"A union A is A and A union with the empty set,"},{"Start":"03:20.150 ","End":"03:22.630","Text":"well it doesn\u0027t add anything to A, just A."},{"Start":"03:22.630 ","End":"03:27.225","Text":"Also note that A is in A union B."},{"Start":"03:27.225 ","End":"03:30.300","Text":"Likewise, B is in A union B is A subset of,"},{"Start":"03:30.300 ","End":"03:32.490","Text":"when I say is in, is a subset of,"},{"Start":"03:32.490 ","End":"03:34.380","Text":"I should be precise."},{"Start":"03:34.380 ","End":"03:40.115","Text":"Also note that if A is a subset of B and I\u0027m going to take A union with B,"},{"Start":"03:40.115 ","End":"03:42.230","Text":"we just get the largest set B,"},{"Start":"03:42.230 ","End":"03:45.980","Text":"adding A which is a subset doesn\u0027t increase B any further."},{"Start":"03:45.980 ","End":"03:48.775","Text":"Continuing with operations on sets,"},{"Start":"03:48.775 ","End":"03:50.980","Text":"the next one is Intersection,"},{"Start":"03:50.980 ","End":"03:54.395","Text":"also a binary operation on sets."},{"Start":"03:54.395 ","End":"03:55.880","Text":"You intersect 2 sets,"},{"Start":"03:55.880 ","End":"03:58.655","Text":"get another set, and I\u0027ll start with an example."},{"Start":"03:58.655 ","End":"04:00.650","Text":"Suppose A contains 1, 2,"},{"Start":"04:00.650 ","End":"04:03.810","Text":"and 10, and B consists of 1, 2,"},{"Start":"04:03.810 ","End":"04:05.055","Text":"3 and 4,"},{"Start":"04:05.055 ","End":"04:10.955","Text":"then we define a new set containing the elements that are both in A and in B."},{"Start":"04:10.955 ","End":"04:14.380","Text":"If we look here, we see that 1 and 2 are in common,"},{"Start":"04:14.380 ","End":"04:20.870","Text":"and that\u0027s what we call the intersection of A and B, denoted like so."},{"Start":"04:20.870 ","End":"04:24.515","Text":"Sometimes union is called cap and this is called cap."},{"Start":"04:24.515 ","End":"04:26.120","Text":"In our example, like I said,"},{"Start":"04:26.120 ","End":"04:28.280","Text":"the intersection consists of 1 and 2,"},{"Start":"04:28.280 ","End":"04:31.300","Text":"they\u0027re the only 1s that belong to both."},{"Start":"04:31.300 ","End":"04:40.190","Text":"Formally, we define the intersection as a set of all x such that x is in A and x is in"},{"Start":"04:40.190 ","End":"04:44.640","Text":"B. X belongs to the intersection of"},{"Start":"04:44.640 ","End":"04:50.340","Text":"A and B if and only if it belongs to A and belongs to B."},{"Start":"04:50.950 ","End":"04:57.545","Text":"Some more examples, the intersection of a set with itself is itself."},{"Start":"04:57.545 ","End":"04:59.975","Text":"Here we took the example 1, 2, 3,"},{"Start":"04:59.975 ","End":"05:03.770","Text":"something that belongs here and here, 1, 2 and 3."},{"Start":"05:03.770 ","End":"05:06.350","Text":"1, 4, 10 intersection 1, 4,"},{"Start":"05:06.350 ","End":"05:09.005","Text":"11, it\u0027s just the 1 and the 4."},{"Start":"05:09.005 ","End":"05:13.410","Text":"We take 1, 4, 10, intersect 11,"},{"Start":"05:13.410 ","End":"05:17.675","Text":"that\u0027s the empty set when there is nothing that\u0027s in both this and this,"},{"Start":"05:17.675 ","End":"05:19.805","Text":"then the intersection is empty."},{"Start":"05:19.805 ","End":"05:22.310","Text":"Now an example with intervals,"},{"Start":"05:22.310 ","End":"05:26.300","Text":"the closed interval from 1-5 intersection."},{"Start":"05:26.300 ","End":"05:31.250","Text":"The closed interval from 4 to 7 is the close interval from 4 to 5."},{"Start":"05:31.250 ","End":"05:32.915","Text":"And you can see this with a picture."},{"Start":"05:32.915 ","End":"05:35.615","Text":"We have from 1 to 5 inclusive."},{"Start":"05:35.615 ","End":"05:40.565","Text":"It\u0027s solid dots here from 4 to 7, and the overlap,"},{"Start":"05:40.565 ","End":"05:43.340","Text":"like if you superimpose the shadow of both of them,"},{"Start":"05:43.340 ","End":"05:47.225","Text":"then it covers from 4 to 5 inclusive."},{"Start":"05:47.225 ","End":"05:50.900","Text":"Now some properties of the intersection L,"},{"Start":"05:50.900 ","End":"05:56.165","Text":"it\u0027s commutative intersection of A with B or B with A, same thing."},{"Start":"05:56.165 ","End":"05:58.130","Text":"Also, it\u0027s associative."},{"Start":"05:58.130 ","End":"06:01.905","Text":"You intersect A with B and then intersect that with C,"},{"Start":"06:01.905 ","End":"06:05.480","Text":"the same as intersecting A with the result of B intersection"},{"Start":"06:05.480 ","End":"06:11.450","Text":"C. Intersection of A with itself is itself."},{"Start":"06:11.450 ","End":"06:14.090","Text":"We had an example of that here."},{"Start":"06:14.090 ","End":"06:18.500","Text":"The intersection of A with the empty set is the empty set"},{"Start":"06:18.500 ","End":"06:20.960","Text":"because there is nothing that\u0027s both in A and"},{"Start":"06:20.960 ","End":"06:23.510","Text":"in the empty set because there\u0027s nothing in the empty set."},{"Start":"06:23.510 ","End":"06:27.425","Text":"The intersection of A with B is a subset of A."},{"Start":"06:27.425 ","End":"06:29.135","Text":"If it\u0027s in A and in B,"},{"Start":"06:29.135 ","End":"06:30.865","Text":"in particular it\u0027s in A."},{"Start":"06:30.865 ","End":"06:34.520","Text":"Similarly A intersection B is contained in B,"},{"Start":"06:34.520 ","End":"06:37.055","Text":"and if A is a subset of B,"},{"Start":"06:37.055 ","End":"06:39.960","Text":"then the intersection of A with B is just A."},{"Start":"06:39.960 ","End":"06:41.720","Text":"Because if x is in A,"},{"Start":"06:41.720 ","End":"06:42.950","Text":"it\u0027s automatically in B."},{"Start":"06:42.950 ","End":"06:44.780","Text":"It doesn\u0027t add or subtract anything."},{"Start":"06:44.780 ","End":"06:48.020","Text":"Now that we\u0027ve mentioned both union and intersection,"},{"Start":"06:48.020 ","End":"06:50.930","Text":"there are some properties that combine both."},{"Start":"06:50.930 ","End":"06:56.780","Text":"There\u0027s distributivity of intersection over union."},{"Start":"06:56.780 ","End":"07:01.880","Text":"If you think of intersection like multiplication and the union-like addition,"},{"Start":"07:01.880 ","End":"07:05.150","Text":"it\u0027s like A times B plus C is AB plus AC."},{"Start":"07:05.150 ","End":"07:07.955","Text":"Well, similar with intersection and union."},{"Start":"07:07.955 ","End":"07:11.975","Text":"Funny thing is, unlike addition and multiplication,"},{"Start":"07:11.975 ","End":"07:14.060","Text":"if you reverse it, if we think of union like"},{"Start":"07:14.060 ","End":"07:16.535","Text":"multiplication and intersection like addition,"},{"Start":"07:16.535 ","End":"07:18.370","Text":"we get another distributive law,"},{"Start":"07:18.370 ","End":"07:21.900","Text":"A union B intersection C is A union B intersection"},{"Start":"07:21.900 ","End":"07:26.390","Text":"A union C. This doesn\u0027t work in arithmetic with addition and multiplication."},{"Start":"07:26.390 ","End":"07:28.870","Text":"Anyway, that\u0027s it for this clip,"},{"Start":"07:28.870 ","End":"07:34.050","Text":"and we\u0027ll continue in future clips with more operations on sets."}],"ID":26565},{"Watched":false,"Name":"Exercise 1","Duration":"3m 16s","ChapterTopicVideoID":25764,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"In this exercise, we\u0027re given several sets,"},{"Start":"00:03.555 ","End":"00:05.265","Text":"A, B, C, and D,"},{"Start":"00:05.265 ","End":"00:08.310","Text":"and we have a bunch of expressions,"},{"Start":"00:08.310 ","End":"00:12.960","Text":"5 of them, we have to compute them involving union and intersection."},{"Start":"00:12.960 ","End":"00:16.080","Text":"Let\u0027s, first of all, copy the sets here."},{"Start":"00:16.080 ","End":"00:17.670","Text":"Now, I\u0027ll take them one at a time."},{"Start":"00:17.670 ","End":"00:20.055","Text":"First one, A union B,"},{"Start":"00:20.055 ","End":"00:22.515","Text":"this union with this."},{"Start":"00:22.515 ","End":"00:25.050","Text":"We could take all of the set A,"},{"Start":"00:25.050 ","End":"00:27.855","Text":"3, 4, 5, 6, 7, 8, 9,"},{"Start":"00:27.855 ","End":"00:30.330","Text":"and put that here,"},{"Start":"00:30.330 ","End":"00:34.770","Text":"and then let\u0027s go to the next set for the duplicate,"},{"Start":"00:34.770 ","End":"00:39.540","Text":"6 duplicate, 8 duplicate, 10."},{"Start":"00:39.540 ","End":"00:41.475","Text":"We can add that."},{"Start":"00:41.475 ","End":"00:44.865","Text":"Number 2, A intersection B,"},{"Start":"00:44.865 ","End":"00:49.140","Text":"this intersect, the same 2 sets just intersection."},{"Start":"00:49.140 ","End":"00:52.605","Text":"Let\u0027s see, 3 is not in the other set, so skip that."},{"Start":"00:52.605 ","End":"00:55.230","Text":"4, we have it here and here,"},{"Start":"00:55.230 ","End":"00:57.240","Text":"so we can put it in the intersection."},{"Start":"00:57.240 ","End":"01:02.150","Text":"5, nope. 6, we have it here and here."},{"Start":"01:02.150 ","End":"01:03.980","Text":"So it goes in the intersection,"},{"Start":"01:03.980 ","End":"01:05.750","Text":"7 is not here,"},{"Start":"01:05.750 ","End":"01:08.540","Text":"8, that\u0027s good. That\u0027s here."},{"Start":"01:08.540 ","End":"01:11.485","Text":"So it goes here, 9, not here."},{"Start":"01:11.485 ","End":"01:13.580","Text":"We just have to go through the first set,"},{"Start":"01:13.580 ","End":"01:15.485","Text":"and see which is also in the second yes."},{"Start":"01:15.485 ","End":"01:18.590","Text":"4, 6, 8, that\u0027s number 2."},{"Start":"01:18.590 ","End":"01:21.215","Text":"Number 3, A union B,"},{"Start":"01:21.215 ","End":"01:26.615","Text":"intersection C. We\u0027ve already computed A union B here, so that\u0027s this."},{"Start":"01:26.615 ","End":"01:29.950","Text":"The set C is 3, 5, 7, 9."},{"Start":"01:29.950 ","End":"01:33.045","Text":"3 is in here."},{"Start":"01:33.045 ","End":"01:35.855","Text":"So in the intersection 4 no,"},{"Start":"01:35.855 ","End":"01:39.320","Text":"5 is here and here, so yes."},{"Start":"01:39.320 ","End":"01:42.810","Text":"6, nope. 7 is in both."},{"Start":"01:42.810 ","End":"01:45.675","Text":"So in the intersection 8, not."},{"Start":"01:45.675 ","End":"01:49.080","Text":"9, yes, and here."},{"Start":"01:49.080 ","End":"01:52.455","Text":"Next, Number 4, I\u0027ve colored it."},{"Start":"01:52.455 ","End":"01:56.045","Text":"You can see that we can use the distributive law on it."},{"Start":"01:56.045 ","End":"01:58.490","Text":"We don\u0027t have to, but I think it\u0027s a shortcut."},{"Start":"01:58.490 ","End":"02:01.975","Text":"See, we have a B union and a B union."},{"Start":"02:01.975 ","End":"02:05.090","Text":"We can take that out of the brackets and we\u0027re left with"},{"Start":"02:05.090 ","End":"02:10.040","Text":"C intersection D. Now, B is this,"},{"Start":"02:10.040 ","End":"02:12.560","Text":"and C intersection D, this is C,"},{"Start":"02:12.560 ","End":"02:16.235","Text":"this is D. We\u0027ll compute the intersection first,"},{"Start":"02:16.235 ","End":"02:18.290","Text":"and the intersection will have what\u0027s in common,"},{"Start":"02:18.290 ","End":"02:21.185","Text":"and the only thing in common is the 7."},{"Start":"02:21.185 ","End":"02:24.875","Text":"We have this union with just 7."},{"Start":"02:24.875 ","End":"02:27.060","Text":"Let\u0027s just throw the 7,"},{"Start":"02:27.060 ","End":"02:28.070","Text":"and it\u0027s not here already,"},{"Start":"02:28.070 ","End":"02:29.810","Text":"so we add it and we get this."},{"Start":"02:29.810 ","End":"02:31.175","Text":"That\u0027s number 4,"},{"Start":"02:31.175 ","End":"02:32.510","Text":"you have one more."},{"Start":"02:32.510 ","End":"02:36.430","Text":"Number 5, this will also going to use the distributive law on this."},{"Start":"02:36.430 ","End":"02:39.830","Text":"The other distributive intersection over union."},{"Start":"02:39.830 ","End":"02:42.140","Text":"We have a B intersection and B intersection."},{"Start":"02:42.140 ","End":"02:46.100","Text":"It starts with B intersection and then we have the C union with the"},{"Start":"02:46.100 ","End":"02:50.925","Text":"D. C union D is this union with this."},{"Start":"02:50.925 ","End":"02:52.370","Text":"For the union, we\u0027ll have 3,"},{"Start":"02:52.370 ","End":"02:53.675","Text":"5, 7, and 9."},{"Start":"02:53.675 ","End":"02:55.175","Text":"Let\u0027s see what else."},{"Start":"02:55.175 ","End":"02:56.760","Text":"We need a 6."},{"Start":"02:56.760 ","End":"02:58.155","Text":"7 is a duplicate,"},{"Start":"02:58.155 ","End":"02:59.280","Text":"we don\u0027t have that,"},{"Start":"02:59.280 ","End":"03:00.405","Text":"and an 8,"},{"Start":"03:00.405 ","End":"03:01.760","Text":"and we just put them in order."},{"Start":"03:01.760 ","End":"03:04.685","Text":"You don\u0027t have to, but it\u0027s nice to do that."},{"Start":"03:04.685 ","End":"03:07.895","Text":"Now the intersection of these 2, well,"},{"Start":"03:07.895 ","End":"03:12.200","Text":"we can see that the only things in common are 6 and 8."},{"Start":"03:12.200 ","End":"03:16.860","Text":"This is the answer to number 5, and we\u0027re done."}],"ID":26568},{"Watched":false,"Name":"Exercise 2","Duration":"3m 12s","ChapterTopicVideoID":25765,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"This exercise, we\u0027re given 4 sets,"},{"Start":"00:03.090 ","End":"00:04.515","Text":"A, B, C, D,"},{"Start":"00:04.515 ","End":"00:08.370","Text":"and we have 5 expressions that we need to compute,"},{"Start":"00:08.370 ","End":"00:11.895","Text":"all of them involving union and intersection,"},{"Start":"00:11.895 ","End":"00:15.375","Text":"and the sets mostly involve intervals."},{"Start":"00:15.375 ","End":"00:20.880","Text":"Let\u0027s, first of all, rewrite the 4 sets of some of them."},{"Start":"00:20.880 ","End":"00:23.265","Text":"A and B as is,"},{"Start":"00:23.265 ","End":"00:26.100","Text":"but C, we can write as an interval."},{"Start":"00:26.100 ","End":"00:29.265","Text":"It\u0027s just exactly the interval from 0-4,"},{"Start":"00:29.265 ","End":"00:32.160","Text":"the closed interval including 0 and 4,"},{"Start":"00:32.160 ","End":"00:35.295","Text":"and D, if you think about it as the empty set,"},{"Start":"00:35.295 ","End":"00:38.385","Text":"set of all x such that 2 to the x is 0,"},{"Start":"00:38.385 ","End":"00:41.390","Text":"but there is no solution to the equation 2 to"},{"Start":"00:41.390 ","End":"00:44.765","Text":"the x equals 0 because 2 to the anything is positive,"},{"Start":"00:44.765 ","End":"00:46.840","Text":"so yeah, it\u0027s the empty set."},{"Start":"00:46.840 ","End":"00:49.845","Text":"Now, the first expression, A union B."},{"Start":"00:49.845 ","End":"00:52.560","Text":"A is this, B is this."},{"Start":"00:52.560 ","End":"00:54.885","Text":"I\u0027ll show you the diagram."},{"Start":"00:54.885 ","End":"01:03.840","Text":"From 1-4, including the 1 and here we have from minus 2-1 and if we take the union,"},{"Start":"01:03.840 ","End":"01:06.465","Text":"they seamlessly join up,"},{"Start":"01:06.465 ","End":"01:10.860","Text":"and we get continuously from minus 2-4,"},{"Start":"01:10.860 ","End":"01:13.185","Text":"not including the endpoints,"},{"Start":"01:13.185 ","End":"01:16.320","Text":"so this is minus 2-4, the interval,"},{"Start":"01:16.320 ","End":"01:19.920","Text":"and the next 1 is the same 2 sets,"},{"Start":"01:19.920 ","End":"01:25.890","Text":"but the intersection, and they don\u0027t have any overlap, they just missed."},{"Start":"01:25.890 ","End":"01:27.990","Text":"This 1 goes up to 1, but not including,"},{"Start":"01:27.990 ","End":"01:30.120","Text":"and then from 1 onwards, so there is no overlap,"},{"Start":"01:30.120 ","End":"01:32.115","Text":"so that\u0027s the empty set."},{"Start":"01:32.115 ","End":"01:35.670","Text":"Next 1, A union B intersection C. Well,"},{"Start":"01:35.670 ","End":"01:37.650","Text":"we already have A union B,"},{"Start":"01:37.650 ","End":"01:40.020","Text":"it\u0027s minus 2 comma 4,"},{"Start":"01:40.020 ","End":"01:44.250","Text":"and C from here is closed set 0, 4."},{"Start":"01:44.250 ","End":"01:46.380","Text":"We don\u0027t need a sketch if you think about it."},{"Start":"01:46.380 ","End":"01:50.145","Text":"Here, we\u0027re going from minus 2-4, but when we hit the 0,"},{"Start":"01:50.145 ","End":"01:51.975","Text":"we pick up an overlap,"},{"Start":"01:51.975 ","End":"01:54.350","Text":"and we overlap all the way up to,"},{"Start":"01:54.350 ","End":"01:55.940","Text":"but not including the 4,"},{"Start":"01:55.940 ","End":"01:58.805","Text":"so here we have from 0-4,"},{"Start":"01:58.805 ","End":"02:01.460","Text":"half-open, closed on the 0 side,"},{"Start":"02:01.460 ","End":"02:02.870","Text":"open on the 4 side."},{"Start":"02:02.870 ","End":"02:06.035","Text":"Next 1, we\u0027re going to use the distributive law."},{"Start":"02:06.035 ","End":"02:10.040","Text":"Don\u0027t have to, but it\u0027ll be a bit simpler if we do use it."},{"Start":"02:10.040 ","End":"02:12.895","Text":"We have a B union and a B union,"},{"Start":"02:12.895 ","End":"02:14.940","Text":"so we can just write B union once,"},{"Start":"02:14.940 ","End":"02:17.660","Text":"and then we have a C intersection D,"},{"Start":"02:17.660 ","End":"02:19.880","Text":"C intersection D. Now,"},{"Start":"02:19.880 ","End":"02:21.955","Text":"D is the empty set."},{"Start":"02:21.955 ","End":"02:25.730","Text":"C intersection with the empty set is also the empty set,"},{"Start":"02:25.730 ","End":"02:29.350","Text":"and B union with the empty set is just B,"},{"Start":"02:29.350 ","End":"02:33.445","Text":"and B happens to be minus 2 comma 1,"},{"Start":"02:33.445 ","End":"02:35.600","Text":"and that\u0027s the answer."},{"Start":"02:35.600 ","End":"02:38.030","Text":"The next 1 is also distributive law,"},{"Start":"02:38.030 ","End":"02:39.245","Text":"but the other way."},{"Start":"02:39.245 ","End":"02:41.140","Text":"B intersection and B intersection,"},{"Start":"02:41.140 ","End":"02:42.820","Text":"so we write B intersection, and then,"},{"Start":"02:42.820 ","End":"02:45.070","Text":"here, C union D,"},{"Start":"02:45.070 ","End":"02:48.540","Text":"C union D. D is the empty set,"},{"Start":"02:48.540 ","End":"02:50.520","Text":"so C union D is C,"},{"Start":"02:50.520 ","End":"02:53.790","Text":"so we need B intersection C. This is B."},{"Start":"02:53.790 ","End":"02:55.455","Text":"This is C,"},{"Start":"02:55.455 ","End":"03:00.090","Text":"and it\u0027s similar to this 1 here."},{"Start":"03:00.090 ","End":"03:03.470","Text":"Minus 2-1, but when we get to 0,"},{"Start":"03:03.470 ","End":"03:04.670","Text":"we have some overlap,"},{"Start":"03:04.670 ","End":"03:07.070","Text":"and the overlap continues until we get to 1,"},{"Start":"03:07.070 ","End":"03:12.660","Text":"and it\u0027s half-open interval from 0-1, and we\u0027re done."}],"ID":26569},{"Watched":false,"Name":"Set difference","Duration":"3m 51s","ChapterTopicVideoID":25759,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.195","Text":"Continuing with operations on sets,"},{"Start":"00:03.195 ","End":"00:07.815","Text":"we\u0027re going to learn about another binary operation called set difference,"},{"Start":"00:07.815 ","End":"00:09.525","Text":"the difference of 2 sets."},{"Start":"00:09.525 ","End":"00:11.835","Text":"Let\u0027s say we have A and B."},{"Start":"00:11.835 ","End":"00:16.560","Text":"We define A minus B is a set of elements which are"},{"Start":"00:16.560 ","End":"00:21.535","Text":"in A but not in B and we can formally define it as A minus B,"},{"Start":"00:21.535 ","End":"00:27.795","Text":"the set of all x such that x belongs to A but x does not belong to B,"},{"Start":"00:27.795 ","End":"00:30.360","Text":"it\u0027s an element of A but not of B."},{"Start":"00:30.360 ","End":"00:32.250","Text":"Let\u0027s see some examples."},{"Start":"00:32.250 ","End":"00:33.900","Text":"Here we have 1, 2, 3, 4,"},{"Start":"00:33.900 ","End":"00:37.020","Text":"5, take away the set 1,4,10."},{"Start":"00:37.020 ","End":"00:41.705","Text":"Let\u0027s go through the elements of the first set and see which is not in the second set."},{"Start":"00:41.705 ","End":"00:44.785","Text":"1 is here, but it\u0027s also here, so no good."},{"Start":"00:44.785 ","End":"00:48.750","Text":"2 is in here and not in here, so that\u0027s good."},{"Start":"00:48.750 ","End":"00:52.500","Text":"3 is in this one and not in this one."},{"Start":"00:52.500 ","End":"00:54.450","Text":"4 is in this one,"},{"Start":"00:54.450 ","End":"00:56.625","Text":"but it is in this one, so no good,"},{"Start":"00:56.625 ","End":"00:58.700","Text":"and 5 is in the first,"},{"Start":"00:58.700 ","End":"00:59.810","Text":"but not in the second,"},{"Start":"00:59.810 ","End":"01:02.090","Text":"so we get 2,3,5."},{"Start":"01:02.090 ","End":"01:03.890","Text":"Now another example."},{"Start":"01:03.890 ","End":"01:06.155","Text":"This minus this,"},{"Start":"01:06.155 ","End":"01:08.240","Text":"1 is also in there,"},{"Start":"01:08.240 ","End":"01:10.295","Text":"4 is good,"},{"Start":"01:10.295 ","End":"01:12.905","Text":"10 is not good because it\u0027s also in here."},{"Start":"01:12.905 ","End":"01:15.560","Text":"14 is also in here,"},{"Start":"01:15.560 ","End":"01:18.320","Text":"24 that\u0027s in here,"},{"Start":"01:18.320 ","End":"01:21.415","Text":"but not in here, and 41 is in both, so no good."},{"Start":"01:21.415 ","End":"01:25.040","Text":"We just have the 4 and the 24."},{"Start":"01:25.040 ","End":"01:28.140","Text":"Next, we have an empty set here."},{"Start":"01:28.140 ","End":"01:31.370","Text":"We want all the elements which are in here and not in the empty set."},{"Start":"01:31.370 ","End":"01:33.260","Text":"Well, no element is in the empty set,"},{"Start":"01:33.260 ","End":"01:35.300","Text":"so we just take the ones here."},{"Start":"01:35.300 ","End":"01:39.860","Text":"We can generalize any set minus the empty set is itself."},{"Start":"01:39.860 ","End":"01:44.180","Text":"Another interesting thing is if we take set minus itself,"},{"Start":"01:44.180 ","End":"01:49.460","Text":"we get the empty set because you want all the elements which are in A but not in A."},{"Start":"01:49.460 ","End":"01:52.995","Text":"Well, there are none. So empty set,"},{"Start":"01:52.995 ","End":"01:54.960","Text":"and what happens if we take the empty set and"},{"Start":"01:54.960 ","End":"01:56.910","Text":"try to take the set difference with another set?"},{"Start":"01:56.910 ","End":"02:01.145","Text":"We want all the elements which are in the empty set and not in this set."},{"Start":"02:01.145 ","End":"02:03.425","Text":"Well, there are no elements in the empty set."},{"Start":"02:03.425 ","End":"02:06.305","Text":"There\u0027s nothing that satisfies these two conditions."},{"Start":"02:06.305 ","End":"02:09.185","Text":"Now let\u0027s take an example with intervals,"},{"Start":"02:09.185 ","End":"02:16.340","Text":"1,4 minus 2,10 and picture could help from 1 to 4,"},{"Start":"02:16.340 ","End":"02:19.790","Text":"including the 1 and we take away,"},{"Start":"02:19.790 ","End":"02:22.985","Text":"remove everything that\u0027s in the interval from 2 to 10."},{"Start":"02:22.985 ","End":"02:27.145","Text":"What we remove is just the bit from here to here,"},{"Start":"02:27.145 ","End":"02:30.555","Text":"we don\u0027t remove the 2 because the 2 is not in there."},{"Start":"02:30.555 ","End":"02:34.240","Text":"What\u0027s left is from 1 to 2, including the 2."},{"Start":"02:34.240 ","End":"02:36.185","Text":"That\u0027s it for examples."},{"Start":"02:36.185 ","End":"02:39.095","Text":"Now some properties of the set difference."},{"Start":"02:39.095 ","End":"02:45.530","Text":"A set minus the empty set is itself and we saw an example of that here."},{"Start":"02:45.530 ","End":"02:48.950","Text":"A set minus itself is the empty set."},{"Start":"02:48.950 ","End":"02:50.480","Text":"Here\u0027s an example,"},{"Start":"02:50.480 ","End":"02:55.670","Text":"and the empty set minus any set is still the empty set, like here."},{"Start":"02:55.670 ","End":"02:58.025","Text":"Now a remark about notation."},{"Start":"02:58.025 ","End":"03:01.760","Text":"The set difference instead of the minus is sometimes denoted"},{"Start":"03:01.760 ","End":"03:06.350","Text":"with a backslash, a diagonal line."},{"Start":"03:06.350 ","End":"03:11.460","Text":"You often see A minus B this way,"},{"Start":"03:11.460 ","End":"03:16.235","Text":"and I\u0027ll give an example using this notation, examples for calculus."},{"Start":"03:16.235 ","End":"03:20.330","Text":"We could say that the domain of the function f of x is 1 over"},{"Start":"03:20.330 ","End":"03:24.500","Text":"x minus 4 is all the reals except for 4."},{"Start":"03:24.500 ","End":"03:28.050","Text":"We could write it as R set difference with"},{"Start":"03:28.050 ","End":"03:32.525","Text":"4 everything that\u0027s in the reals but is not in the set 4,"},{"Start":"03:32.525 ","End":"03:34.565","Text":"meaning is not equal to 4."},{"Start":"03:34.565 ","End":"03:39.215","Text":"Another example, function, 1 over x squared minus x."},{"Start":"03:39.215 ","End":"03:43.070","Text":"You can see that the denominator is 0 when x is 1 or 0,"},{"Start":"03:43.070 ","End":"03:48.980","Text":"so the domain is all reals except for the numbers in the set 0,1."},{"Start":"03:48.980 ","End":"03:51.990","Text":"That\u0027s all for this clip."}],"ID":26563},{"Watched":false,"Name":"Exercise 3","Duration":"1m 54s","ChapterTopicVideoID":25766,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this exercise, we\u0027re given 3 sets as follows."},{"Start":"00:04.380 ","End":"00:07.470","Text":"We have to compute, first of all,"},{"Start":"00:07.470 ","End":"00:10.440","Text":"A minus B minus C,"},{"Start":"00:10.440 ","End":"00:15.030","Text":"and then A minus B minus C. Can tell you now will get"},{"Start":"00:15.030 ","End":"00:20.820","Text":"2 different answers which will show that set difference is not an associative operation,"},{"Start":"00:20.820 ","End":"00:24.705","Text":"a bit like the minus in arithmetic is not associative."},{"Start":"00:24.705 ","End":"00:31.110","Text":"Start, A minus B bracket minus C. It\u0027s just copying the sets,"},{"Start":"00:31.110 ","End":"00:33.105","Text":"this minus this, minus this."},{"Start":"00:33.105 ","End":"00:35.180","Text":"We first compute what\u0027s in the brackets."},{"Start":"00:35.180 ","End":"00:38.930","Text":"This minus this is just the 4 that\u0027s in the first,"},{"Start":"00:38.930 ","End":"00:41.360","Text":"but not in the second 5 in the second,"},{"Start":"00:41.360 ","End":"00:42.560","Text":"6 is in the second,"},{"Start":"00:42.560 ","End":"00:43.745","Text":"7 is in a second,"},{"Start":"00:43.745 ","End":"00:44.945","Text":"8 is in a second,"},{"Start":"00:44.945 ","End":"00:50.485","Text":"so nothing more to take from here except the 4 and this one as is."},{"Start":"00:50.485 ","End":"00:54.890","Text":"Then we want all the elements which are in here but not in here."},{"Start":"00:54.890 ","End":"00:58.670","Text":"There is nothing, the only one in here is 4 and it is in the second one,"},{"Start":"00:58.670 ","End":"01:00.720","Text":"so nothing in here that\u0027s not in here."},{"Start":"01:00.720 ","End":"01:02.840","Text":"The answer is the empty set."},{"Start":"01:02.840 ","End":"01:05.930","Text":"Now, number 2, where the brackets are different,"},{"Start":"01:05.930 ","End":"01:08.855","Text":"here we first want to compute the B minus C,"},{"Start":"01:08.855 ","End":"01:10.520","Text":"so the brackets here."},{"Start":"01:10.520 ","End":"01:13.745","Text":"We\u0027ll do this set difference with this first."},{"Start":"01:13.745 ","End":"01:15.360","Text":"Let\u0027s go over these,"},{"Start":"01:15.360 ","End":"01:16.995","Text":"5 is in here,"},{"Start":"01:16.995 ","End":"01:18.960","Text":"6 is in here,"},{"Start":"01:18.960 ","End":"01:21.180","Text":"7, that\u0027s good it\u0027s in here,"},{"Start":"01:21.180 ","End":"01:24.465","Text":"but not in here, the 8, yeah,"},{"Start":"01:24.465 ","End":"01:25.680","Text":"also good in here,"},{"Start":"01:25.680 ","End":"01:27.045","Text":"but not in here,"},{"Start":"01:27.045 ","End":"01:29.055","Text":"and the 9 is good."},{"Start":"01:29.055 ","End":"01:31.894","Text":"From this we get the 7, 8, 9."},{"Start":"01:31.894 ","End":"01:34.160","Text":"Now we want the set difference of this with this."},{"Start":"01:34.160 ","End":"01:36.065","Text":"Let\u0027s see, the 4 is good,"},{"Start":"01:36.065 ","End":"01:38.555","Text":"not in here, 5 is good,"},{"Start":"01:38.555 ","End":"01:41.195","Text":"6 is good, 7,"},{"Start":"01:41.195 ","End":"01:42.350","Text":"no because it\u0027s in here,"},{"Start":"01:42.350 ","End":"01:44.240","Text":"and 8, no because it\u0027s in here."},{"Start":"01:44.240 ","End":"01:46.745","Text":"What we\u0027re left with is the 4, 5, 6,"},{"Start":"01:46.745 ","End":"01:49.310","Text":"which is not the same as the empty set,"},{"Start":"01:49.310 ","End":"01:54.660","Text":"which shows that set difference is not associative. Okay, that\u0027s it."}],"ID":26570},{"Watched":false,"Name":"Exercise 4","Duration":"2m 15s","ChapterTopicVideoID":25767,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this exercise, we\u0027ll practice some set operations;"},{"Start":"00:04.290 ","End":"00:05.850","Text":"union, intersection,"},{"Start":"00:05.850 ","End":"00:09.465","Text":"difference, and intervals mostly."},{"Start":"00:09.465 ","End":"00:12.855","Text":"We\u0027ll read each one as we come to it."},{"Start":"00:12.855 ","End":"00:15.787","Text":"First one is a closed interval 1, 10"},{"Start":"00:15.787 ","End":"00:18.780","Text":"union open interval 4, 14."},{"Start":"00:18.780 ","End":"00:20.505","Text":"The picture of that,"},{"Start":"00:20.505 ","End":"00:24.645","Text":"here from 1 to 10 closed and here from 4 to 14."},{"Start":"00:24.645 ","End":"00:29.670","Text":"Now the union is like the shadow of the whole thing."},{"Start":"00:29.670 ","End":"00:33.150","Text":"What we get is from 1 to 14,"},{"Start":"00:33.150 ","End":"00:35.275","Text":"closed here, open here."},{"Start":"00:35.275 ","End":"00:39.560","Text":"Straightforward. The intersection is the overlap."},{"Start":"00:39.560 ","End":"00:45.105","Text":"The overlap is from 4 to 10, like so."},{"Start":"00:45.105 ","End":"00:49.055","Text":"The set difference, this one minus this one,"},{"Start":"00:49.055 ","End":"00:51.995","Text":"it\u0027s everything in here that\u0027s not in here."},{"Start":"00:51.995 ","End":"00:54.025","Text":"We remove this part."},{"Start":"00:54.025 ","End":"00:56.535","Text":"We\u0027re left with the 4 inside,"},{"Start":"00:56.535 ","End":"00:58.980","Text":"just from 1 to 4 including."},{"Start":"00:58.980 ","End":"01:01.885","Text":"Next time we\u0027ll do without a diagram,"},{"Start":"01:01.885 ","End":"01:07.180","Text":"going all the way down from minus infinity up to 4, including 4."},{"Start":"01:07.180 ","End":"01:11.145","Text":"Where does this overlap with 1 to infinity?"},{"Start":"01:11.145 ","End":"01:13.920","Text":"The overlap part is from 1 to 4."},{"Start":"01:13.920 ","End":"01:15.540","Text":"It doesn\u0027t include the 1,"},{"Start":"01:15.540 ","End":"01:17.820","Text":"it doesn\u0027t include the 4."},{"Start":"01:17.820 ","End":"01:20.115","Text":"That\u0027s what we have."},{"Start":"01:20.115 ","End":"01:22.480","Text":"Next one is in 2 parts,"},{"Start":"01:22.480 ","End":"01:25.070","Text":"We have a union and then an intersection."},{"Start":"01:25.070 ","End":"01:27.140","Text":"Well, these 2 sets, 0,"},{"Start":"01:27.140 ","End":"01:28.820","Text":"1 union 4, 7,"},{"Start":"01:28.820 ","End":"01:30.290","Text":"they have no overlap."},{"Start":"01:30.290 ","End":"01:32.360","Text":"So we can\u0027t simplify the union,"},{"Start":"01:32.360 ","End":"01:33.995","Text":"we leave it as is."},{"Start":"01:33.995 ","End":"01:39.035","Text":"But what does this union have in common with the interval from 1 to 4?"},{"Start":"01:39.035 ","End":"01:42.230","Text":"Well, it just hits it at the right end point here."},{"Start":"01:42.230 ","End":"01:45.020","Text":"This one, it hits just at the left end point."},{"Start":"01:45.020 ","End":"01:47.950","Text":"All we\u0027re left with is 2 points,"},{"Start":"01:47.950 ","End":"01:50.445","Text":"1 and 4, themselves."},{"Start":"01:50.445 ","End":"01:54.700","Text":"Now this one, we have an intersection and another intersection,"},{"Start":"01:54.700 ","End":"01:56.000","Text":"doesn\u0027t matter in what order."},{"Start":"01:56.000 ","End":"01:58.430","Text":"But let\u0027s take these 2 first, 0,"},{"Start":"01:58.430 ","End":"01:59.570","Text":"1 intersection 1,"},{"Start":"01:59.570 ","End":"02:01.535","Text":"4, is just the end point 1."},{"Start":"02:01.535 ","End":"02:03.569","Text":"1 is the only point in common"},{"Start":"02:03.569 ","End":"02:08.940","Text":"between these two and the 0.1 intersection with 2,"},{"Start":"02:08.940 ","End":"02:10.500","Text":"4 turn to have nothing in common,"},{"Start":"02:10.500 ","End":"02:11.970","Text":"because one isn\u0027t in here."},{"Start":"02:11.970 ","End":"02:16.210","Text":"What we get here is the empty set. That\u0027s it."}],"ID":26571},{"Watched":false,"Name":"Complement of a Set","Duration":"4m 35s","ChapterTopicVideoID":25763,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"Returning to operations on sets,"},{"Start":"00:03.645 ","End":"00:05.640","Text":"we\u0027re going to introduce fourth 1."},{"Start":"00:05.640 ","End":"00:07.050","Text":"So far we\u0027ve had union,"},{"Start":"00:07.050 ","End":"00:08.985","Text":"intersection, and difference."},{"Start":"00:08.985 ","End":"00:12.705","Text":"Now we\u0027re going to have something called complement of a set."},{"Start":"00:12.705 ","End":"00:16.725","Text":"But for this, we need a certain universal set,"},{"Start":"00:16.725 ","End":"00:19.635","Text":"which was once called the universe of discourse."},{"Start":"00:19.635 ","End":"00:25.185","Text":"Going to assume that all the sets are subsets of some giant universal set."},{"Start":"00:25.185 ","End":"00:30.135","Text":"Like with numbers, it might be all the real numbers or we\u0027re talking about people,"},{"Start":"00:30.135 ","End":"00:32.850","Text":"might be the set of all people that ever existed."},{"Start":"00:32.850 ","End":"00:35.145","Text":"For a given discussion,"},{"Start":"00:35.145 ","End":"00:40.315","Text":"any set which we mentioned will satisfy that A is a subset of U,"},{"Start":"00:40.315 ","End":"00:42.560","Text":"although U is often not mentioned,"},{"Start":"00:42.560 ","End":"00:45.835","Text":"it\u0027s just implicitly understood."},{"Start":"00:45.835 ","End":"00:48.260","Text":"I\u0027m going to describe the complements in a moment,"},{"Start":"00:48.260 ","End":"00:50.405","Text":"but first a picture to tell you what it is."},{"Start":"00:50.405 ","End":"00:53.985","Text":"If this is the universe and this is a set A,"},{"Start":"00:53.985 ","End":"00:57.770","Text":"and everything not in A will be in A complement."},{"Start":"00:57.770 ","End":"00:59.839","Text":"The complement is a unary operation,"},{"Start":"00:59.839 ","End":"01:03.515","Text":"unlike the other 3 which are binary operations."},{"Start":"01:03.515 ","End":"01:06.080","Text":"Anyway, like I said, the set difference,"},{"Start":"01:06.080 ","End":"01:13.955","Text":"everything that\u0027s in U but not in A is the complement of A and denoted as a complement."},{"Start":"01:13.955 ","End":"01:19.325","Text":"It\u0027s also called the complement of A relative to you."},{"Start":"01:19.325 ","End":"01:24.035","Text":"There are other notations that you might see."},{"Start":"01:24.035 ","End":"01:26.465","Text":"Common 1 is A bar."},{"Start":"01:26.465 ","End":"01:30.770","Text":"Another common 1 is A prime for the complement."},{"Start":"01:30.770 ","End":"01:34.070","Text":"You should be aware of the alternative notations."},{"Start":"01:34.070 ","End":"01:42.680","Text":"Now note that x is in the complement of A if and only if x is not in A,"},{"Start":"01:42.680 ","End":"01:45.260","Text":"but x is still in U, that universal set."},{"Start":"01:45.260 ","End":"01:52.760","Text":"We could rewrite this as x belongs to A if and only if x not in the complement of A,"},{"Start":"01:52.760 ","End":"01:54.470","Text":"it doesn\u0027t matter which you take as the set,"},{"Start":"01:54.470 ","End":"01:55.805","Text":"which is the complement."},{"Start":"01:55.805 ","End":"01:57.875","Text":"This is also true from logic."},{"Start":"01:57.875 ","End":"02:00.410","Text":"If 2 propositions are equivalent,"},{"Start":"02:00.410 ","End":"02:02.840","Text":"if you negate both of them, it\u0027s still equivalent."},{"Start":"02:02.840 ","End":"02:05.990","Text":"From this, we can deduce that"},{"Start":"02:05.990 ","End":"02:10.519","Text":"the complement of the complement of a set is the set itself."},{"Start":"02:10.519 ","End":"02:12.800","Text":"The very short proof for that,"},{"Start":"02:12.800 ","End":"02:17.120","Text":"if x is in the complement of the complement,"},{"Start":"02:17.120 ","End":"02:20.880","Text":"then using this replacing A with A complement,"},{"Start":"02:20.880 ","End":"02:23.400","Text":"then x is not in A complement."},{"Start":"02:23.400 ","End":"02:24.855","Text":"But now looking at this,"},{"Start":"02:24.855 ","End":"02:31.230","Text":"x not in A complement is equivalent to x is in A. X is in A complement,"},{"Start":"02:31.230 ","End":"02:33.210","Text":"complement if and only if x is in A,"},{"Start":"02:33.210 ","End":"02:36.100","Text":"which means that these 2 sets are equal."},{"Start":"02:36.100 ","End":"02:39.510","Text":"Next, something called De Morgan\u0027s laws."},{"Start":"02:39.510 ","End":"02:42.740","Text":"De Morgan\u0027s laws for sets say that"},{"Start":"02:42.740 ","End":"02:48.340","Text":"the complement of the union is the intersection of the complements."},{"Start":"02:48.340 ","End":"02:54.060","Text":"Conversely, the complement of the intersection is the union of the complements."},{"Start":"02:54.060 ","End":"02:55.620","Text":"I\u0027ll show you a diagram,"},{"Start":"02:55.620 ","End":"02:58.320","Text":"although we haven\u0027t studied Venn diagrams."},{"Start":"02:58.320 ","End":"03:02.950","Text":"Many of you have an intuitive understanding of it."},{"Start":"03:02.950 ","End":"03:05.265","Text":"Say the first 1,"},{"Start":"03:05.265 ","End":"03:06.420","Text":"A is in yellow,"},{"Start":"03:06.420 ","End":"03:10.399","Text":"B is in yellow, and the union is everything in yellow or this orange."},{"Start":"03:10.399 ","End":"03:14.830","Text":"The complement of that is all the blue."},{"Start":"03:14.830 ","End":"03:22.700","Text":"All of the blue is equivalent to whatever\u0027s outside A and outside B,"},{"Start":"03:22.700 ","End":"03:25.630","Text":"if it\u0027s outside both of them and this is what it is."},{"Start":"03:25.630 ","End":"03:26.990","Text":"On the other hand,"},{"Start":"03:26.990 ","End":"03:29.585","Text":"if I take the complement of the intersection,"},{"Start":"03:29.585 ","End":"03:32.420","Text":"the intersection is yellow and the complement,"},{"Start":"03:32.420 ","End":"03:36.560","Text":"everything that\u0027s blue, both shadings and that\u0027s"},{"Start":"03:36.560 ","End":"03:41.735","Text":"equivalent to everything that\u0027s either outside A or outside B, or both."},{"Start":"03:41.735 ","End":"03:45.815","Text":"For example, a point here is outside both of them,"},{"Start":"03:45.815 ","End":"03:48.565","Text":"a point here is outside B,"},{"Start":"03:48.565 ","End":"03:50.860","Text":"a point here is outside A."},{"Start":"03:50.860 ","End":"03:53.870","Text":"All these are in this set,"},{"Start":"03:53.870 ","End":"03:55.010","Text":"which is, like I said,"},{"Start":"03:55.010 ","End":"04:00.355","Text":"either A intersection B complement or the union of the complements."},{"Start":"04:00.355 ","End":"04:03.830","Text":"De Morgan\u0027s laws can actually be proved and there will be proved in"},{"Start":"04:03.830 ","End":"04:07.550","Text":"the exercise based on the analogous laws for logic,"},{"Start":"04:07.550 ","End":"04:10.264","Text":"set theory and logic are closely related."},{"Start":"04:10.264 ","End":"04:14.150","Text":"The union corresponds to or intersection"},{"Start":"04:14.150 ","End":"04:18.930","Text":"corresponds to and complement corresponds to naught."},{"Start":"04:18.930 ","End":"04:20.840","Text":"Most set theory statements,"},{"Start":"04:20.840 ","End":"04:22.400","Text":"if you do this translation,"},{"Start":"04:22.400 ","End":"04:24.860","Text":"you get a logic statement and vice versa."},{"Start":"04:24.860 ","End":"04:27.710","Text":"These are the De Morgan\u0027s laws for logic,"},{"Start":"04:27.710 ","End":"04:29.150","Text":"I won\u0027t read them out."},{"Start":"04:29.150 ","End":"04:31.280","Text":"Using this, you can prove this,"},{"Start":"04:31.280 ","End":"04:33.725","Text":"as I say, it\u0027s in the exercises."},{"Start":"04:33.725 ","End":"04:36.510","Text":"That\u0027s all for this clip."}],"ID":26567},{"Watched":false,"Name":"Exercise 5","Duration":"2m 30s","ChapterTopicVideoID":25768,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.190","Text":"In this exercise, the universal set is a set of integers from 11-18,"},{"Start":"00:08.190 ","End":"00:11.340","Text":"and here are sets A and B, 12,"},{"Start":"00:11.340 ","End":"00:14.610","Text":"15, 18, and 13, 15, 17."},{"Start":"00:14.610 ","End":"00:19.920","Text":"Using these sets, we\u0027re going to demonstrate De Morgan\u0027s law that the complement of"},{"Start":"00:19.920 ","End":"00:25.950","Text":"a union B is the intersection of the complement of A with the complement of B."},{"Start":"00:25.950 ","End":"00:30.165","Text":"We need a universal set when we talk about complements."},{"Start":"00:30.165 ","End":"00:33.795","Text":"Let\u0027s compute the left-hand side first,"},{"Start":"00:33.795 ","End":"00:37.020","Text":"and the first step is to compute A union with B."},{"Start":"00:37.020 ","End":"00:40.095","Text":"We can just see this that we need a 12,"},{"Start":"00:40.095 ","End":"00:41.460","Text":"we need a 15,"},{"Start":"00:41.460 ","End":"00:43.225","Text":"we need an 18,"},{"Start":"00:43.225 ","End":"00:46.130","Text":"and then we need the 13,"},{"Start":"00:46.130 ","End":"00:49.860","Text":"15 we have already, and 17."},{"Start":"00:49.860 ","End":"00:51.750","Text":"I think a picture will help."},{"Start":"00:51.750 ","End":"00:55.625","Text":"Even though we haven\u0027t learned Venn diagrams yet,"},{"Start":"00:55.625 ","End":"00:58.870","Text":"most people find these fairly intuitive."},{"Start":"00:58.870 ","End":"01:03.120","Text":"This is U, and we have all the numbers 11,"},{"Start":"01:03.120 ","End":"01:07.575","Text":"12, 13, 14, 15, 16, 17,"},{"Start":"01:07.575 ","End":"01:12.120","Text":"18, and the ones that are in A are 12,"},{"Start":"01:12.120 ","End":"01:18.385","Text":"15, 18, and the ones that are in B are 13, 15, 17."},{"Start":"01:18.385 ","End":"01:23.865","Text":"A union B are these 5 numbers here."},{"Start":"01:23.865 ","End":"01:28.205","Text":"The complement is whatever is outside of these,"},{"Start":"01:28.205 ","End":"01:31.640","Text":"so that would be 11, 14, and 16."},{"Start":"01:31.640 ","End":"01:35.345","Text":"You could go along the universal set and just cross off any"},{"Start":"01:35.345 ","End":"01:39.775","Text":"that are in A union B and whatever is left is the complement."},{"Start":"01:39.775 ","End":"01:42.060","Text":"That\u0027s the left-hand side."},{"Start":"01:42.060 ","End":"01:43.620","Text":"Now, the right-hand side."},{"Start":"01:43.620 ","End":"01:45.765","Text":"First, we need A complement,"},{"Start":"01:45.765 ","End":"01:52.815","Text":"and whatever is outside of A is 11, 13, 14,"},{"Start":"01:52.815 ","End":"01:58.685","Text":"16, 17, and then whatever is outside of B is 11,"},{"Start":"01:58.685 ","End":"02:03.155","Text":"12, 14, 16, 18, like so."},{"Start":"02:03.155 ","End":"02:06.530","Text":"Then the intersection of these at C,"},{"Start":"02:06.530 ","End":"02:10.735","Text":"we have 11 and 11, that\u0027s here,"},{"Start":"02:10.735 ","End":"02:16.055","Text":"14 and 14 gives us 14,"},{"Start":"02:16.055 ","End":"02:20.555","Text":"16 also here, here, and here."},{"Start":"02:20.555 ","End":"02:23.540","Text":"Now, if we look at this and we look at this,"},{"Start":"02:23.540 ","End":"02:25.745","Text":"we see that they really are the same."},{"Start":"02:25.745 ","End":"02:31.170","Text":"This illustrates this De Morgan\u0027s law. We\u0027re done."}],"ID":26572},{"Watched":false,"Name":"Exercise 6","Duration":"3m 58s","ChapterTopicVideoID":25769,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In this exercise, we\u0027re going to prove De Morgan\u0027s first law"},{"Start":"00:04.080 ","End":"00:07.965","Text":"for sets using De Morgan\u0027s law for logic."},{"Start":"00:07.965 ","End":"00:09.660","Text":"In part 2,"},{"Start":"00:09.660 ","End":"00:12.705","Text":"we\u0027re going to show that the first De Morgan\u0027s law"},{"Start":"00:12.705 ","End":"00:16.845","Text":"can be used to prove the second De Morgan\u0027s law for sets."},{"Start":"00:16.845 ","End":"00:20.339","Text":"Here are De Morgan\u0027s laws for sets,"},{"Start":"00:20.339 ","End":"00:23.220","Text":"and here are De Morgan\u0027s laws for logic."},{"Start":"00:23.220 ","End":"00:27.210","Text":"They are very similar if you think about it."},{"Start":"00:27.210 ","End":"00:30.075","Text":"Let\u0027s start with number 1."},{"Start":"00:30.075 ","End":"00:31.575","Text":"We have to show,"},{"Start":"00:31.575 ","End":"00:34.410","Text":"I\u0027m putting it in question mark equals because we haven\u0027t proved it"},{"Start":"00:34.410 ","End":"00:37.230","Text":"yet that the complement of"},{"Start":"00:37.230 ","End":"00:43.440","Text":"a union B is the intersection of A complement with B complement,"},{"Start":"00:43.440 ","End":"00:47.090","Text":"and 1 way of showing that 2 sets are equal is to show that"},{"Start":"00:47.090 ","End":"00:51.380","Text":"an element is in 1 of them if an only if it\u0027s in the other,"},{"Start":"00:51.380 ","End":"00:55.790","Text":"if we have an arrow 1 way it\u0027s set containment."},{"Start":"00:55.790 ","End":"00:58.400","Text":"The arrow the other way is the reverse containment,"},{"Start":"00:58.400 ","End":"01:01.490","Text":"but if we have a double arrow, then it\u0027s equality."},{"Start":"01:01.490 ","End":"01:06.245","Text":"Let\u0027s start with x belongs to the left-hand side,"},{"Start":"01:06.245 ","End":"01:11.390","Text":"and we\u0027ll do steps of if and only if and reach the right-hand side,"},{"Start":"01:11.390 ","End":"01:13.685","Text":"x belongs to the right-hand side."},{"Start":"01:13.685 ","End":"01:18.610","Text":"From this, we can say that x is not in a union B."},{"Start":"01:18.610 ","End":"01:23.035","Text":"Because to belong to the complement means not to belong."},{"Start":"01:23.035 ","End":"01:29.105","Text":"That means that it is not true that x belongs to a union B."},{"Start":"01:29.105 ","End":"01:35.285","Text":"Just interpreting that not a member of is not a member of,"},{"Start":"01:35.285 ","End":"01:38.125","Text":"I guess I should have put brackets after the not."},{"Start":"01:38.125 ","End":"01:46.040","Text":"Anyway, x belongs to A union B by definition means that x belongs to A or x belongs to B."},{"Start":"01:46.040 ","End":"01:48.275","Text":"Then we can say,"},{"Start":"01:48.275 ","End":"01:51.290","Text":"using De Morgan\u0027s law for logic,"},{"Start":"01:51.290 ","End":"01:56.610","Text":"the first 1 that not call this P and this 1 Q,"},{"Start":"01:56.610 ","End":"02:01.395","Text":"not P or Q is not P and not Q."},{"Start":"02:01.395 ","End":"02:03.360","Text":"So we have this."},{"Start":"02:03.360 ","End":"02:06.660","Text":"Now, note x belongs to A,"},{"Start":"02:06.660 ","End":"02:10.060","Text":"is x does not belong to A, not a member of A."},{"Start":"02:10.060 ","End":"02:13.610","Text":"Similarly here, x does not belong to B,"},{"Start":"02:13.610 ","End":"02:16.175","Text":"and if x does not belong to A,"},{"Start":"02:16.175 ","End":"02:19.435","Text":"then it belongs to A complement that\u0027s if only if."},{"Start":"02:19.435 ","End":"02:21.665","Text":"An x does not belong to B,"},{"Start":"02:21.665 ","End":"02:24.800","Text":"if and only if x belongs to B complement."},{"Start":"02:24.800 ","End":"02:29.635","Text":"Now we have x belongs to 1 set and x belongs to another set."},{"Start":"02:29.635 ","End":"02:32.480","Text":"By definition of the intersection,"},{"Start":"02:32.480 ","End":"02:34.265","Text":"x belongs to this,"},{"Start":"02:34.265 ","End":"02:40.125","Text":"and x belongs to this if and only if x belongs to the intersection of this and this,"},{"Start":"02:40.125 ","End":"02:43.895","Text":"and that concludes part 1."},{"Start":"02:43.895 ","End":"02:46.865","Text":"Now on to part 2."},{"Start":"02:46.865 ","End":"02:49.505","Text":"I want to remind you that in general,"},{"Start":"02:49.505 ","End":"02:53.920","Text":"the complement of the complement of a set is the set itself."},{"Start":"02:53.920 ","End":"02:56.550","Text":"Now let\u0027s prove that,"},{"Start":"02:56.550 ","End":"02:57.950","Text":"well, I copied it out here,"},{"Start":"02:57.950 ","End":"02:59.050","Text":"this is what we have to prove,"},{"Start":"02:59.050 ","End":"03:02.450","Text":"so we\u0027re going to start with 1 side and reach the other side."},{"Start":"03:02.450 ","End":"03:09.230","Text":"A intersection B complement is equal to set of A I can write A complement,"},{"Start":"03:09.230 ","End":"03:10.580","Text":"complement and instead of B,"},{"Start":"03:10.580 ","End":"03:12.664","Text":"I can write B complement, complement."},{"Start":"03:12.664 ","End":"03:17.810","Text":"Now what I can do is apply the first De Morgan\u0027s law,"},{"Start":"03:17.810 ","End":"03:24.065","Text":"which is this, and say as follows this complement, it\u0027s just copied."},{"Start":"03:24.065 ","End":"03:27.844","Text":"This A complement takes the place of A in this formula,"},{"Start":"03:27.844 ","End":"03:32.225","Text":"and this B complement takes the place of B in this formula."},{"Start":"03:32.225 ","End":"03:36.890","Text":"What we do is we drop the C here and here and put 1 on the outside."},{"Start":"03:36.890 ","End":"03:39.515","Text":"Where we drop this C and this C,"},{"Start":"03:39.515 ","End":"03:43.030","Text":"and we put them in brackets and put another C here."},{"Start":"03:43.030 ","End":"03:45.240","Text":"That\u0027s using the first law,"},{"Start":"03:45.240 ","End":"03:49.970","Text":"and then all we have to do is apply again the complement of the complement to"},{"Start":"03:49.970 ","End":"03:54.615","Text":"get that this is just what\u0027s inside A complement, union B complement."},{"Start":"03:54.615 ","End":"03:59.160","Text":"We\u0027ve got from here to here, and we\u0027re done."}],"ID":26573},{"Watched":false,"Name":"Exercise 7","Duration":"2m 36s","ChapterTopicVideoID":25770,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.660","Text":"This is a three-part exercise and the parts 2 and 3 will need De Morgan\u0027s laws."},{"Start":"00:06.660 ","End":"00:07.995","Text":"I\u0027ll remind you of them."},{"Start":"00:07.995 ","End":"00:09.810","Text":"Let\u0027s start with part 1,"},{"Start":"00:09.810 ","End":"00:14.685","Text":"the set difference of A and B is A intersection B complement,"},{"Start":"00:14.685 ","End":"00:20.265","Text":"so we\u0027ll show that x is in the left-hand side if and only if x is in the right-hand side."},{"Start":"00:20.265 ","End":"00:23.385","Text":"Let\u0027s say x is in A minus B,"},{"Start":"00:23.385 ","End":"00:29.160","Text":"that means that x is in A and not in B by the definition of the set difference."},{"Start":"00:29.160 ","End":"00:31.680","Text":"Now to say that x is not in B,"},{"Start":"00:31.680 ","End":"00:34.755","Text":"is the same as to say that x is in B complement,"},{"Start":"00:34.755 ","End":"00:38.670","Text":"x is not in B if and only if it\u0027s in B complement."},{"Start":"00:38.990 ","End":"00:45.035","Text":"To say that x is in here and in here is to say that x is in their intersection,"},{"Start":"00:45.035 ","End":"00:48.890","Text":"and that\u0027s the right-hand side so that\u0027s part 1 done."},{"Start":"00:48.890 ","End":"00:50.930","Text":"As I say for part 2 and 3,"},{"Start":"00:50.930 ","End":"00:52.505","Text":"we\u0027ll need the De Morgan\u0027s Laws,"},{"Start":"00:52.505 ","End":"00:54.890","Text":"so here I\u0027ll remind you of them."},{"Start":"00:54.890 ","End":"00:57.335","Text":"Let\u0027s start with number 2,"},{"Start":"00:57.335 ","End":"01:02.680","Text":"C minus A intersection B is equal to,"},{"Start":"01:02.680 ","End":"01:04.800","Text":"by part 1,"},{"Start":"01:04.800 ","End":"01:10.880","Text":"this is equal to C intersection A intersection B complement."},{"Start":"01:10.880 ","End":"01:15.530","Text":"We said that this minus this is this intersection with this complement."},{"Start":"01:15.530 ","End":"01:19.070","Text":"Yes, not A and B it\u0027s C and A intersection B."},{"Start":"01:19.070 ","End":"01:20.960","Text":"Now we\u0027ll use De Morgan\u0027s law,"},{"Start":"01:20.960 ","End":"01:22.925","Text":"the second one on this,"},{"Start":"01:22.925 ","End":"01:26.800","Text":"and we get C intersection A complement union B complement,"},{"Start":"01:26.800 ","End":"01:31.625","Text":"and I will use the distributive law of intersection over union,"},{"Start":"01:31.625 ","End":"01:33.635","Text":"and we get C intersection with this,"},{"Start":"01:33.635 ","End":"01:36.395","Text":"union C intersection with this."},{"Start":"01:36.395 ","End":"01:43.980","Text":"Now, this again using part 1 is C minus A and this is C minus B,"},{"Start":"01:43.980 ","End":"01:46.265","Text":"and this is what we had to show."},{"Start":"01:46.265 ","End":"01:49.650","Text":"Where is it? Yeah, here. That\u0027s good."},{"Start":"01:49.650 ","End":"01:51.675","Text":"Now number 3,"},{"Start":"01:51.675 ","End":"01:54.615","Text":"and here we\u0027ll use the other De Morgan\u0027s law."},{"Start":"01:54.615 ","End":"02:01.260","Text":"First of all, set difference is the intersection with the complement and then"},{"Start":"02:01.260 ","End":"02:08.210","Text":"the complement using this De Morgan\u0027s law gives us A compliment, section B complement."},{"Start":"02:08.210 ","End":"02:11.345","Text":"Now we can just throw the brackets away because"},{"Start":"02:11.345 ","End":"02:14.390","Text":"intersection is associative, not only that,"},{"Start":"02:14.390 ","End":"02:21.100","Text":"we can replace C by C intersection C. You\u0027ll see why I want to do that right away."},{"Start":"02:21.100 ","End":"02:25.900","Text":"I can change the order and say it\u0027s C intersection A complement,"},{"Start":"02:25.900 ","End":"02:28.715","Text":"intersection C intersection B compliment,"},{"Start":"02:28.715 ","End":"02:31.260","Text":"and then this is C minus A,"},{"Start":"02:31.260 ","End":"02:33.480","Text":"this is C minus B,"},{"Start":"02:33.480 ","End":"02:36.990","Text":"and that\u0027s what we needed, so we\u0027re done."}],"ID":26574},{"Watched":false,"Name":"Venn Diagrams","Duration":"1m 29s","ChapterTopicVideoID":25762,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"New topic in set theory, Venn diagrams."},{"Start":"00:03.720 ","End":"00:05.625","Text":"You may have heard of these already."},{"Start":"00:05.625 ","End":"00:11.760","Text":"Venn diagrams are a way of making sets visual to illustrate them graphically."},{"Start":"00:11.760 ","End":"00:16.110","Text":"Typically, we use circles or rectangles,"},{"Start":"00:16.110 ","End":"00:17.520","Text":"ellipses, what have you,"},{"Start":"00:17.520 ","End":"00:19.170","Text":"sometimes a cloudy shape,"},{"Start":"00:19.170 ","End":"00:22.800","Text":"they represent a set; sets A, B,"},{"Start":"00:22.800 ","End":"00:26.330","Text":"and C. If the sets have something in common,"},{"Start":"00:26.330 ","End":"00:30.155","Text":"for example, we could represent them like this with an overlap."},{"Start":"00:30.155 ","End":"00:33.500","Text":"Let\u0027s say that A was the set 1, 2, 3, 4, 5,"},{"Start":"00:33.500 ","End":"00:35.030","Text":"and B was 2, 5, 7,"},{"Start":"00:35.030 ","End":"00:37.340","Text":"8, you\u0027d get something like this."},{"Start":"00:37.340 ","End":"00:40.475","Text":"Where whatever is in the intersection, the overlap,"},{"Start":"00:40.475 ","End":"00:42.170","Text":"which is 2 and 5,"},{"Start":"00:42.170 ","End":"00:43.640","Text":"is in this part here,"},{"Start":"00:43.640 ","End":"00:46.445","Text":"just intuitive, what you\u0027d expect."},{"Start":"00:46.445 ","End":"00:49.340","Text":"This is what we mean when we say Venn diagrams."},{"Start":"00:49.340 ","End":"00:51.500","Text":"They help us describe things"},{"Start":"00:51.500 ","End":"00:55.150","Text":"graphically and you can see things more easily when it\u0027s visual."},{"Start":"00:55.150 ","End":"00:58.350","Text":"They often even help us to prove or disprove claims,"},{"Start":"00:58.350 ","End":"01:03.544","Text":"but you can\u0027t just rely on a diagram to constitute a proof in itself."},{"Start":"01:03.544 ","End":"01:06.055","Text":"There are some examples."},{"Start":"01:06.055 ","End":"01:09.270","Text":"If this is a set A and this is the set B,"},{"Start":"01:09.270 ","End":"01:13.139","Text":"then this would be the set A intersection B,"},{"Start":"01:13.139 ","End":"01:17.135","Text":"A union B, set difference,"},{"Start":"01:17.135 ","End":"01:18.305","Text":"whatever\u0027s in A,"},{"Start":"01:18.305 ","End":"01:19.790","Text":"but not in B,"},{"Start":"01:19.790 ","End":"01:22.115","Text":"and whatever here is in B,"},{"Start":"01:22.115 ","End":"01:23.435","Text":"but not in A."},{"Start":"01:23.435 ","End":"01:24.650","Text":"Those are some examples,"},{"Start":"01:24.650 ","End":"01:29.400","Text":"there\u0027ll be some more in the exercises. That\u0027s it."}],"ID":26566},{"Watched":false,"Name":"Exercise 8","Duration":"2m 33s","ChapterTopicVideoID":25771,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re given 4 sets and we have to"},{"Start":"00:04.320 ","End":"00:09.580","Text":"find the complement relative to the reals."},{"Start":"00:10.220 ","End":"00:12.720","Text":"This is the universal set,"},{"Start":"00:12.720 ","End":"00:17.865","Text":"[inaudible] we have to find the complement of each of these 4 subsets of the reals."},{"Start":"00:17.865 ","End":"00:23.850","Text":"The first one is the infinite interval from 1 including 1 to infinity."},{"Start":"00:23.850 ","End":"00:25.080","Text":"If you think about it,"},{"Start":"00:25.080 ","End":"00:29.760","Text":"all that\u0027s missing is the part from minus infinity up to,"},{"Start":"00:29.760 ","End":"00:31.875","Text":"but not including 1."},{"Start":"00:31.875 ","End":"00:35.415","Text":"The complement is minus infinity to 1."},{"Start":"00:35.415 ","End":"00:38.150","Text":"Part b, we want the set of"},{"Start":"00:38.150 ","End":"00:43.205","Text":"solutions to the following inequality and then the complement of that."},{"Start":"00:43.205 ","End":"00:45.410","Text":"Now, this, if we sketch it,"},{"Start":"00:45.410 ","End":"00:50.570","Text":"is a parabola which hits the axis at 1 and at 4."},{"Start":"00:50.570 ","End":"00:55.080","Text":"The part above it is here and here."},{"Start":"00:55.080 ","End":"00:57.990","Text":"Yeah, it\u0027s x minus 1 x minus 4,"},{"Start":"00:57.990 ","End":"00:59.265","Text":"so we get the 1 and the 4."},{"Start":"00:59.265 ","End":"01:05.070","Text":"Then above the axis is less than 1 or more than 4."},{"Start":"01:05.070 ","End":"01:11.185","Text":"We can write this as the integral of minus infinity to 1 union 4 to infinity."},{"Start":"01:11.185 ","End":"01:12.920","Text":"But we don\u0027t want the set B,"},{"Start":"01:12.920 ","End":"01:14.770","Text":"we want B complement."},{"Start":"01:14.770 ","End":"01:17.945","Text":"The compliment is just stuff that\u0027s missing."},{"Start":"01:17.945 ","End":"01:20.240","Text":"We have everything up to 1 and from 4,"},{"Start":"01:20.240 ","End":"01:22.895","Text":"so we\u0027re missing the bit between 1 and 4."},{"Start":"01:22.895 ","End":"01:26.530","Text":"It includes the 1 and the 4 because they\u0027re not here or here."},{"Start":"01:26.530 ","End":"01:29.500","Text":"That\u0027s the answer to part b."},{"Start":"01:29.990 ","End":"01:35.250","Text":"Well, part c is the same as part b."},{"Start":"01:35.250 ","End":"01:37.970","Text":"This here is what we wrote here."},{"Start":"01:37.970 ","End":"01:39.680","Text":"It\u0027s exactly the same."},{"Start":"01:39.680 ","End":"01:41.210","Text":"C is the same as B,"},{"Start":"01:41.210 ","End":"01:45.800","Text":"so C compliment is also 1,4 closed interval."},{"Start":"01:45.800 ","End":"01:50.555","Text":"The last one is the set where x is bigger than 4,"},{"Start":"01:50.555 ","End":"01:53.120","Text":"or x minus 1 is less than 2."},{"Start":"01:53.120 ","End":"01:56.270","Text":"Now, we can translate x minus 1, say,"},{"Start":"01:56.270 ","End":"01:59.425","Text":"the distance from x to 2 is less than 1,"},{"Start":"01:59.425 ","End":"02:06.055","Text":"or in other words, it\u0027s from 1-3."},{"Start":"02:06.055 ","End":"02:10.010","Text":"The other bit, x bigger than 4 is from 4 to infinity."},{"Start":"02:10.010 ","End":"02:13.880","Text":"We have this union and we want the complement."},{"Start":"02:13.880 ","End":"02:17.825","Text":"Now, it\u0027s all arranged in order 1, 3, 4, infinity."},{"Start":"02:17.825 ","End":"02:24.540","Text":"The missing bits are from minus infinity up to 1, including the 1."},{"Start":"02:24.540 ","End":"02:28.140","Text":"Then the gap in the middle is from 3-4,"},{"Start":"02:28.140 ","End":"02:30.480","Text":"including the 3 and the 4."},{"Start":"02:30.480 ","End":"02:34.060","Text":"That\u0027s the answer to D, and we\u0027re done."}],"ID":26575},{"Watched":false,"Name":"Exercise 9","Duration":"3m 34s","ChapterTopicVideoID":25772,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.765","Text":"In this exercise, we\u0027ll practice using Venn diagrams."},{"Start":"00:03.765 ","End":"00:08.290","Text":"Here we have 9 set operations involving A and B,"},{"Start":"00:08.290 ","End":"00:10.985","Text":"and we\u0027ll illustrate them."},{"Start":"00:10.985 ","End":"00:15.675","Text":"First 1, straightforward A intersection B. I\u0027ll use"},{"Start":"00:15.675 ","End":"00:21.315","Text":"the background of an off-white and green for what we\u0027re looking for."},{"Start":"00:21.315 ","End":"00:24.360","Text":"It\u0027s section of A and B which is this circle,"},{"Start":"00:24.360 ","End":"00:26.610","Text":"with this circle is the overlap,"},{"Start":"00:26.610 ","End":"00:29.790","Text":"and that\u0027s what I\u0027ve colored in green here."},{"Start":"00:29.790 ","End":"00:36.815","Text":"A union B is anything that it\u0027s an A or B or both, and that\u0027s this."},{"Start":"00:36.815 ","End":"00:41.675","Text":"A complement is whatever\u0027s outside of the circle A."},{"Start":"00:41.675 ","End":"00:46.080","Text":"Here\u0027s a circle A. I reverse the coloring."},{"Start":"00:46.080 ","End":"00:47.835","Text":"Instead of A being colored,"},{"Start":"00:47.835 ","End":"00:50.545","Text":"everything but A is colored."},{"Start":"00:50.545 ","End":"00:53.660","Text":"Note that I copied this from somewhere."},{"Start":"00:53.660 ","End":"00:54.770","Text":"It says A prime."},{"Start":"00:54.770 ","End":"00:59.560","Text":"I want to remind you that there are various notations for the complement."},{"Start":"00:59.560 ","End":"01:00.960","Text":"In this course,"},{"Start":"01:00.960 ","End":"01:02.350","Text":"mostly we use the c,"},{"Start":"01:02.350 ","End":"01:04.010","Text":"but some people use prime,"},{"Start":"01:04.010 ","End":"01:05.825","Text":"some people use a bar."},{"Start":"01:05.825 ","End":"01:07.640","Text":"Yes, just notation."},{"Start":"01:07.640 ","End":"01:13.084","Text":"Also, I should mention that u is the universal set whenever we talk about complements,"},{"Start":"01:13.084 ","End":"01:17.485","Text":"it\u0027s always relative to some universal set. Let\u0027s continue."},{"Start":"01:17.485 ","End":"01:19.750","Text":"A intersection with B prime."},{"Start":"01:19.750 ","End":"01:24.590","Text":"Anything that\u0027s in A and in the complement of B,"},{"Start":"01:24.590 ","End":"01:26.945","Text":"needs it in A and not in B."},{"Start":"01:26.945 ","End":"01:29.210","Text":"In A and not in B,"},{"Start":"01:29.210 ","End":"01:32.795","Text":"its everything its in A except the parts that are in B."},{"Start":"01:32.795 ","End":"01:39.905","Text":"We cut that part out and then reverse in the complement of A,"},{"Start":"01:39.905 ","End":"01:42.680","Text":"meaning not in A and in B."},{"Start":"01:42.680 ","End":"01:47.140","Text":"Not in A, so A is removed and it is in B so this,"},{"Start":"01:47.140 ","End":"01:52.755","Text":"and then A union with B complement."},{"Start":"01:52.755 ","End":"01:57.215","Text":"Everything in A union,"},{"Start":"01:57.215 ","End":"01:59.600","Text":"everything that\u0027s outside of B."},{"Start":"01:59.600 ","End":"02:01.970","Text":"We don\u0027t take away anything."},{"Start":"02:01.970 ","End":"02:03.170","Text":"We still have everything it\u0027s in A,"},{"Start":"02:03.170 ","End":"02:06.010","Text":"but we add anything that\u0027s not in B."},{"Start":"02:06.010 ","End":"02:11.030","Text":"Then A compliment union B is just a symmetrical,"},{"Start":"02:11.030 ","End":"02:13.460","Text":"just replacing A with B and B with A,"},{"Start":"02:13.460 ","End":"02:15.185","Text":"so it\u0027s very similar."},{"Start":"02:15.185 ","End":"02:19.460","Text":"The last 2 offer demonstrating DeMorgan\u0027s law."},{"Start":"02:19.460 ","End":"02:24.140","Text":"There\u0027s 2 of them. We\u0027ve actually seen this when I talked about DeMorgan\u0027s law."},{"Start":"02:24.140 ","End":"02:27.170","Text":"But anyway, let\u0027s just look at the picture."},{"Start":"02:27.170 ","End":"02:28.910","Text":"When I look at it in 2 ways,"},{"Start":"02:28.910 ","End":"02:30.320","Text":"on the 1 hand,"},{"Start":"02:30.320 ","End":"02:33.545","Text":"we can say it\u0027s everything outside of A,"},{"Start":"02:33.545 ","End":"02:36.955","Text":"which is everything outside of this circle,"},{"Start":"02:36.955 ","End":"02:42.045","Text":"union, anything that\u0027s outside the circle B."},{"Start":"02:42.045 ","End":"02:45.770","Text":"Anything that\u0027s outside either 1 of them is good and the only thing it"},{"Start":"02:45.770 ","End":"02:49.490","Text":"doesn\u0027t get colored is something that\u0027s inside both of them,"},{"Start":"02:49.490 ","End":"02:51.895","Text":"then it can\u0027t be outside this or this."},{"Start":"02:51.895 ","End":"02:54.795","Text":"It\u0027s a same if you look at it as"},{"Start":"02:54.795 ","End":"03:01.250","Text":"A intersection B complement because A intersection B is this part that\u0027s not colored."},{"Start":"03:01.250 ","End":"03:03.140","Text":"If you reverse the coloring then,"},{"Start":"03:03.140 ","End":"03:06.550","Text":"that if coloring the overlap like here,"},{"Start":"03:06.550 ","End":"03:09.765","Text":"the complement of this is here."},{"Start":"03:09.765 ","End":"03:14.640","Text":"The last 1 is a reverse of this picture."},{"Start":"03:14.640 ","End":"03:16.850","Text":"If I look at A union B complement,"},{"Start":"03:16.850 ","End":"03:19.310","Text":"it\u0027s everything that\u0027s not here, so reverse."},{"Start":"03:19.310 ","End":"03:24.364","Text":"But it\u0027s also equal to the intersection of A complement with B complement,"},{"Start":"03:24.364 ","End":"03:28.430","Text":"everything that\u0027s outside of A and outside of B."},{"Start":"03:28.430 ","End":"03:32.415","Text":"The points that are outside of both are all these."},{"Start":"03:32.415 ","End":"03:35.320","Text":"That\u0027s the last 1 and we\u0027re done."}],"ID":26576},{"Watched":false,"Name":"The Power Set","Duration":"4m 21s","ChapterTopicVideoID":25760,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"Now, we come to a new concept in set theory."},{"Start":"00:03.390 ","End":"00:07.665","Text":"The concept of a power Set of a given set."},{"Start":"00:07.665 ","End":"00:13.230","Text":"The set of all subsets of a given set A is called the power set of"},{"Start":"00:13.230 ","End":"00:18.600","Text":"A and is denoted by P of A. I repeat the important part,"},{"Start":"00:18.600 ","End":"00:21.285","Text":"the set of all subsets."},{"Start":"00:21.285 ","End":"00:23.580","Text":"Now, let\u0027s do some examples."},{"Start":"00:23.580 ","End":"00:28.305","Text":"Let\u0027s take the set A to be the set containing 1, 2, and 3."},{"Start":"00:28.305 ","End":"00:31.965","Text":"What is P of A or the subsets?"},{"Start":"00:31.965 ","End":"00:34.230","Text":"Let\u0027s see which subsets we have."},{"Start":"00:34.230 ","End":"00:37.125","Text":"Now, there\u0027s always the empty subset."},{"Start":"00:37.125 ","End":"00:40.230","Text":"The empty set is a subset of every set."},{"Start":"00:40.230 ","End":"00:42.345","Text":"That\u0027s always in there."},{"Start":"00:42.345 ","End":"00:45.575","Text":"Then, we have singleton sets."},{"Start":"00:45.575 ","End":"00:47.495","Text":"Singleton sets contain just 1 element."},{"Start":"00:47.495 ","End":"00:48.830","Text":"For 1, 2, 3,"},{"Start":"00:48.830 ","End":"00:50.420","Text":"we have the set containing 1,"},{"Start":"00:50.420 ","End":"00:51.950","Text":"the singleton set for 2,"},{"Start":"00:51.950 ","End":"00:55.135","Text":"and the singleton set containing 3,"},{"Start":"00:55.135 ","End":"00:57.205","Text":"and maybe take all pairs,"},{"Start":"00:57.205 ","End":"00:59.660","Text":"and we can take the set itself."},{"Start":"00:59.660 ","End":"01:02.210","Text":"You\u0027re always guaranteed to have the empty set,"},{"Start":"01:02.210 ","End":"01:05.035","Text":"and the set itself as subsets."},{"Start":"01:05.035 ","End":"01:07.250","Text":"That\u0027s the first example."},{"Start":"01:07.250 ","End":"01:08.825","Text":"Let\u0027s take another example."},{"Start":"01:08.825 ","End":"01:12.095","Text":"Set A contains just little A and B."},{"Start":"01:12.095 ","End":"01:17.180","Text":"Once again, we can start with the empty set and end with the whole set itself,"},{"Start":"01:17.180 ","End":"01:20.435","Text":"and then, we have the singleton set for A and for B."},{"Start":"01:20.435 ","End":"01:23.975","Text":"Next example, A contains just 1 element,"},{"Start":"01:23.975 ","End":"01:26.180","Text":"1, which subsets does it have?"},{"Start":"01:26.180 ","End":"01:28.015","Text":"Well, has the empty set,"},{"Start":"01:28.015 ","End":"01:29.905","Text":"and it has the set itself."},{"Start":"01:29.905 ","End":"01:34.400","Text":"Finally, let\u0027s see what happens if we take A to be the empty set."},{"Start":"01:34.400 ","End":"01:37.085","Text":"Turns out it has just 1 subset,"},{"Start":"01:37.085 ","End":"01:38.510","Text":"which is the empty set,"},{"Start":"01:38.510 ","End":"01:41.750","Text":"so P of A contains 1 element,"},{"Start":"01:41.750 ","End":"01:44.345","Text":"and that element is the empty set."},{"Start":"01:44.345 ","End":"01:45.740","Text":"It\u0027s not the same thing."},{"Start":"01:45.740 ","End":"01:47.180","Text":"This contains no elements."},{"Start":"01:47.180 ","End":"01:48.770","Text":"This contains 1 element."},{"Start":"01:48.770 ","End":"01:51.050","Text":"It contains the empty set as an element."},{"Start":"01:51.050 ","End":"01:53.695","Text":"This is like, I don\u0027t know, an empty matchbox."},{"Start":"01:53.695 ","End":"01:56.900","Text":"This is a matchbox containing an empty matchbox."},{"Start":"01:56.900 ","End":"02:00.560","Text":"Not the same thing or a Computer Science."},{"Start":"02:00.560 ","End":"02:04.774","Text":"An empty folder or a folder containing an empty folder."},{"Start":"02:04.774 ","End":"02:06.295","Text":"I think you see what I mean?"},{"Start":"02:06.295 ","End":"02:09.140","Text":"Now, notice something A had 3 elements."},{"Start":"02:09.140 ","End":"02:11.060","Text":"How many elements were there in P of A,"},{"Start":"02:11.060 ","End":"02:14.420","Text":"1, 2, 3, 4, 5, 6, 7, 8,"},{"Start":"02:14.420 ","End":"02:16.810","Text":"and there\u0027s a rule that,"},{"Start":"02:16.810 ","End":"02:19.025","Text":"if A is a finite set,"},{"Start":"02:19.025 ","End":"02:21.920","Text":"then the number of elements in the power set"},{"Start":"02:21.920 ","End":"02:24.905","Text":"is 2 to the power of a number of elements in the set,"},{"Start":"02:24.905 ","End":"02:28.675","Text":"just like here, 2^3 is 8."},{"Start":"02:28.675 ","End":"02:31.910","Text":"Yeah, so just writing that out, and here,"},{"Start":"02:31.910 ","End":"02:32.930","Text":"if we do the counting,"},{"Start":"02:32.930 ","End":"02:36.365","Text":"we have 2 here and 4 here."},{"Start":"02:36.365 ","End":"02:39.305","Text":"Here we have 1 and 2,"},{"Start":"02:39.305 ","End":"02:41.630","Text":"and here we have 0 and 1,"},{"Start":"02:41.630 ","End":"02:43.505","Text":"so it all works out."},{"Start":"02:43.505 ","End":"02:45.230","Text":"2^3 is 8, 2^2 is 4,"},{"Start":"02:45.230 ","End":"02:47.945","Text":"2^1 is 2, 2^0 is 1."},{"Start":"02:47.945 ","End":"02:50.030","Text":"For infinite sets, you can say that if"},{"Start":"02:50.030 ","End":"02:53.075","Text":"the set is infinite and the power set is infinite,"},{"Start":"02:53.075 ","End":"02:57.560","Text":"but I\u0027m not sure if 2 to the infinity equals infinity. Yes and no."},{"Start":"02:57.560 ","End":"02:58.830","Text":"Like I mentioned, in set theory,"},{"Start":"02:58.830 ","End":"03:00.320","Text":"there\u0027s different sizes of infinity,"},{"Start":"03:00.320 ","End":"03:03.065","Text":"but we\u0027re not going to get into that in this course."},{"Start":"03:03.065 ","End":"03:04.940","Text":"Let\u0027s continue with some remarks."},{"Start":"03:04.940 ","End":"03:07.175","Text":"First of all, note that,"},{"Start":"03:07.175 ","End":"03:11.690","Text":"if an element x belongs to P of A,"},{"Start":"03:11.690 ","End":"03:18.075","Text":"that\u0027s if and only if x is a subset of A because P of A is the set of all subsets,"},{"Start":"03:18.075 ","End":"03:21.980","Text":"so something belongs to P and A if and only if it\u0027s a subset of A."},{"Start":"03:21.980 ","End":"03:23.390","Text":"That\u0027s by definition."},{"Start":"03:23.390 ","End":"03:25.475","Text":"Suppose we could look at an example."},{"Start":"03:25.475 ","End":"03:28.380","Text":"For example, let\u0027s say,"},{"Start":"03:28.380 ","End":"03:30.060","Text":"the set 2, 3."},{"Start":"03:30.060 ","End":"03:33.840","Text":"2, 3 is an element of P of A."},{"Start":"03:33.840 ","End":"03:36.630","Text":"2, 3 is like our x here, and 2,"},{"Start":"03:36.630 ","End":"03:39.030","Text":"3 is a subset of A,"},{"Start":"03:39.030 ","End":"03:40.710","Text":"so it\u0027s an element of P of A,"},{"Start":"03:40.710 ","End":"03:42.525","Text":"but a subset of A."},{"Start":"03:42.525 ","End":"03:47.820","Text":"The other thing to note is that x is"},{"Start":"03:47.820 ","End":"03:53.435","Text":"an element of A if and only if the singleton x is a subset of A."},{"Start":"03:53.435 ","End":"03:55.325","Text":"We look back here,"},{"Start":"03:55.325 ","End":"03:58.070","Text":"2 is an element of A,"},{"Start":"03:58.070 ","End":"04:02.680","Text":"and 2 embraces is a subset of A,"},{"Start":"04:02.680 ","End":"04:05.720","Text":"but it\u0027s also an element of P of A,"},{"Start":"04:05.720 ","End":"04:07.960","Text":"which is what the next thing is."},{"Start":"04:07.960 ","End":"04:10.189","Text":"These 3 things are equivalent."},{"Start":"04:10.189 ","End":"04:12.605","Text":"X is an element of A,"},{"Start":"04:12.605 ","End":"04:15.290","Text":"the singleton x is a subset of A,"},{"Start":"04:15.290 ","End":"04:19.725","Text":"and the singleton x is a member element of P of A."},{"Start":"04:19.725 ","End":"04:22.440","Text":"That\u0027s it for this clip."}],"ID":26564},{"Watched":false,"Name":"Exercise 10","Duration":"1m 1s","ChapterTopicVideoID":25773,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this exercise, we\u0027re given the set A with 3 elements,"},{"Start":"00:04.440 ","End":"00:06.615","Text":"the empty set 4,"},{"Start":"00:06.615 ","End":"00:10.710","Text":"and the set containing 4 and we have to find the power set of A,"},{"Start":"00:10.710 ","End":"00:12.810","Text":"the set of all subsets."},{"Start":"00:12.810 ","End":"00:16.020","Text":"We actually know how many there\u0027s going to be because"},{"Start":"00:16.020 ","End":"00:18.930","Text":"this has 3 elements so the power set will have 2 to the 3,"},{"Start":"00:18.930 ","End":"00:20.610","Text":"which equals 8 elements so we can"},{"Start":"00:20.610 ","End":"00:23.910","Text":"always use that as a check if we\u0027ve got the right number."},{"Start":"00:23.910 ","End":"00:26.090","Text":"You start off with the empty set,"},{"Start":"00:26.090 ","End":"00:30.845","Text":"that\u0027s always a subset of any set so it\u0027s a member of the power set."},{"Start":"00:30.845 ","End":"00:33.380","Text":"Then we could take singleton sets,"},{"Start":"00:33.380 ","End":"00:36.110","Text":"the set containing the empty set,"},{"Start":"00:36.110 ","End":"00:38.120","Text":"the set containing 4,"},{"Start":"00:38.120 ","End":"00:41.135","Text":"and the set containing the set containing 4."},{"Start":"00:41.135 ","End":"00:47.030","Text":"We could work on all pairs this with this and so on so we have empty set with 4,"},{"Start":"00:47.030 ","End":"00:49.280","Text":"4 with the set containing 4,"},{"Start":"00:49.280 ","End":"00:52.000","Text":"and the empty set with the set containing 4."},{"Start":"00:52.000 ","End":"00:54.180","Text":"Then we can take all 3 of them,"},{"Start":"00:54.180 ","End":"00:57.330","Text":"which is the whole set A itself, which is this."},{"Start":"00:57.330 ","End":"01:00.330","Text":"That\u0027s the answer."},{"Start":"01:00.330 ","End":"01:02.830","Text":"That\u0027s it."}],"ID":26577},{"Watched":false,"Name":"Exercise 11","Duration":"1m 44s","ChapterTopicVideoID":25774,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.085","Text":"In this exercise, we have 2 statements that we have to prove or disprove."},{"Start":"00:05.085 ","End":"00:08.295","Text":"Disprove means find a counterexample."},{"Start":"00:08.295 ","End":"00:11.115","Text":"Part a says that for every set A,"},{"Start":"00:11.115 ","End":"00:14.370","Text":"A is a subset of P of A, the power set."},{"Start":"00:14.370 ","End":"00:16.440","Text":"Part b says the opposite,"},{"Start":"00:16.440 ","End":"00:18.195","Text":"that for every set A,"},{"Start":"00:18.195 ","End":"00:21.180","Text":"A is not a subset of P of A."},{"Start":"00:21.180 ","End":"00:22.950","Text":"But take each one separately."},{"Start":"00:22.950 ","End":"00:25.305","Text":"First of all, let\u0027s take the first one."},{"Start":"00:25.305 ","End":"00:29.610","Text":"Is it always true that A is a subset of P of A?"},{"Start":"00:29.610 ","End":"00:31.515","Text":"It turns out this is false,"},{"Start":"00:31.515 ","End":"00:33.705","Text":"and here is a counterexample."},{"Start":"00:33.705 ","End":"00:37.140","Text":"Let\u0027s take A to be the set containing just 1."},{"Start":"00:37.140 ","End":"00:41.475","Text":"Then the power set of a contains 2 elements,"},{"Start":"00:41.475 ","End":"00:43.260","Text":"2 subsets of A;"},{"Start":"00:43.260 ","End":"00:47.130","Text":"one is the empty set and one is the set A itself."},{"Start":"00:47.130 ","End":"00:51.435","Text":"However, A is not a subset of P of A"},{"Start":"00:51.435 ","End":"00:56.660","Text":"because one is an element of A but one is not an element of P of A."},{"Start":"00:56.660 ","End":"00:59.245","Text":"This is not the same as 1."},{"Start":"00:59.245 ","End":"01:05.060","Text":"So A is not a subset of P of A since we have an element in A that\u0027s not in P of A."},{"Start":"01:05.060 ","End":"01:07.790","Text":"Now, b is also false."},{"Start":"01:07.790 ","End":"01:09.770","Text":"By the way, don\u0027t think that one or the other has to"},{"Start":"01:09.770 ","End":"01:12.290","Text":"be true just because they\u0027re opposites,"},{"Start":"01:12.290 ","End":"01:14.565","Text":"because it\u0027s for every."},{"Start":"01:14.565 ","End":"01:17.720","Text":"It could be that sometimes this is true and sometimes this is true,"},{"Start":"01:17.720 ","End":"01:20.285","Text":"but neither of them is always true."},{"Start":"01:20.285 ","End":"01:22.130","Text":"The second one is false also,"},{"Start":"01:22.130 ","End":"01:23.975","Text":"and here\u0027s a counterexample."},{"Start":"01:23.975 ","End":"01:26.245","Text":"Take A to be the empty set."},{"Start":"01:26.245 ","End":"01:29.600","Text":"Now, the empty set is a subset of every set."},{"Start":"01:29.600 ","End":"01:31.740","Text":"Doesn\u0027t really matter what P of A is,"},{"Start":"01:31.740 ","End":"01:33.050","Text":"you don\u0027t have to compute it."},{"Start":"01:33.050 ","End":"01:35.930","Text":"You know that whatever P of A is,"},{"Start":"01:35.930 ","End":"01:38.495","Text":"the empty set is a subset of it."},{"Start":"01:38.495 ","End":"01:41.750","Text":"So of course this can be true sometimes."},{"Start":"01:41.750 ","End":"01:45.360","Text":"That concludes this exercise."}],"ID":26578},{"Watched":false,"Name":"Exercise 12","Duration":"1m 14s","ChapterTopicVideoID":25775,"CourseChapterTopicPlaylistID":199,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this exercise, we have to prove that if A is a subset of B,"},{"Start":"00:05.580 ","End":"00:10.500","Text":"then the power set of A is a subset of the power set of B."},{"Start":"00:10.500 ","End":"00:16.920","Text":"What we need to show is that if X is in P of A,"},{"Start":"00:16.920 ","End":"00:19.470","Text":"then X is also in P of B."},{"Start":"00:19.470 ","End":"00:23.250","Text":"Now what does it mean for X to be in the power set of A?"},{"Start":"00:23.250 ","End":"00:25.725","Text":"The power set of A is the set of all subsets."},{"Start":"00:25.725 ","End":"00:29.025","Text":"So for X to be in the power set of A is the same thing,"},{"Start":"00:29.025 ","End":"00:32.650","Text":"this is actually if and only if X is a subset of A."},{"Start":"00:32.650 ","End":"00:37.674","Text":"If X is a subset of A and A is a subset of B,"},{"Start":"00:37.674 ","End":"00:41.390","Text":"then by the transitivity of the subset relation,"},{"Start":"00:41.390 ","End":"00:43.790","Text":"X is also a subset of B,"},{"Start":"00:43.790 ","End":"00:45.965","Text":"this is by transitivity."},{"Start":"00:45.965 ","End":"00:47.560","Text":"Again, X is a subset of A,"},{"Start":"00:47.560 ","End":"00:48.830","Text":"A is a subset of B,"},{"Start":"00:48.830 ","End":"00:50.180","Text":"so X is a subset of B."},{"Start":"00:50.180 ","End":"00:53.210","Text":"We proved that I think in 1 of the exercises."},{"Start":"00:53.210 ","End":"00:55.490","Text":"If X is a subset of B,"},{"Start":"00:55.490 ","End":"00:57.860","Text":"then it has a defining property of P"},{"Start":"00:57.860 ","End":"01:01.940","Text":"of B. P of B is the set of all subsets of B and X is 1 of them,"},{"Start":"01:01.940 ","End":"01:03.845","Text":"so it\u0027s in P of B."},{"Start":"01:03.845 ","End":"01:11.300","Text":"We got from here to here and this implies that P of A is a subset of P of B."},{"Start":"01:11.300 ","End":"01:13.370","Text":"We\u0027ve proved what we had to prove,"},{"Start":"01:13.370 ","End":"01:15.630","Text":"and we are done."}],"ID":26579}],"Thumbnail":null,"ID":199},{"Name":"Irrational Numbers","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Preface","Duration":"1m 41s","ChapterTopicVideoID":25800,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/25800.jpeg","UploadDate":"2021-06-23T15:47:12.5070000","DurationForVideoObject":"PT1M41S","Description":null,"MetaTitle":"Preface: Video + Workbook | Proprep","MetaDescription":"Logic, Set Theory, Number System - Irrational Numbers. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/logic%2c-set-theory%2c-number-system/irrational-numbers/vid26604","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"This clip is about the irrational numbers, the set,"},{"Start":"00:04.665 ","End":"00:06.615","Text":"sometimes it\u0027s called P,"},{"Start":"00:06.615 ","End":"00:09.000","Text":"and we touched upon it before,"},{"Start":"00:09.000 ","End":"00:11.219","Text":"but let\u0027s look at it again."},{"Start":"00:11.219 ","End":"00:15.045","Text":"The previous clip, which we call special sets of numbers."},{"Start":"00:15.045 ","End":"00:17.475","Text":"We talked a bit about irrational numbers."},{"Start":"00:17.475 ","End":"00:20.250","Text":"We talked about rationals, irrationals,"},{"Start":"00:20.250 ","End":"00:25.230","Text":"integers, natural numbers, real numbers, and so on."},{"Start":"00:25.230 ","End":"00:30.865","Text":"We said that an irrational number is one which can\u0027t be expressed as a fraction."},{"Start":"00:30.865 ","End":"00:33.890","Text":"We gave some examples of famous irrational numbers,"},{"Start":"00:33.890 ","End":"00:36.155","Text":"the square root of 2, e,"},{"Start":"00:36.155 ","End":"00:37.550","Text":"pi, phi,"},{"Start":"00:37.550 ","End":"00:39.475","Text":"the golden ratio,"},{"Start":"00:39.475 ","End":"00:45.259","Text":"and often or sometimes in the first homework assignment on irrational numbers,"},{"Start":"00:45.259 ","End":"00:50.315","Text":"students like you are asked to prove that various numbers are irrational."},{"Start":"00:50.315 ","End":"00:54.140","Text":"For example, square root of 2 is irrational, root 3,"},{"Start":"00:54.140 ","End":"00:56.690","Text":"root 2 plus root 3, the cube root of 2,"},{"Start":"00:56.690 ","End":"00:58.715","Text":"or the cube root of 4, and so on."},{"Start":"00:58.715 ","End":"01:02.600","Text":"Now, in order to prove that these are actually irrational,"},{"Start":"01:02.600 ","End":"01:05.690","Text":"we\u0027ll need some simple results from number theory."},{"Start":"01:05.690 ","End":"01:08.720","Text":"That\u0027s a branch of mathematics, and for some reason,"},{"Start":"01:08.720 ","End":"01:11.090","Text":"there are presumed to be self-evident even though,"},{"Start":"01:11.090 ","End":"01:13.445","Text":"in my opinion, they\u0027re not self-evident."},{"Start":"01:13.445 ","End":"01:15.995","Text":"Here\u0027s an example of such a claim."},{"Start":"01:15.995 ","End":"01:19.145","Text":"If the square of a natural number is even,"},{"Start":"01:19.145 ","End":"01:20.990","Text":"then the number itself is even."},{"Start":"01:20.990 ","End":"01:22.790","Text":"Maybe it is, maybe it isn\u0027t self-evident,"},{"Start":"01:22.790 ","End":"01:24.190","Text":"but I think it isn\u0027t."},{"Start":"01:24.190 ","End":"01:27.500","Text":"Its thing will prove in the following clips,"},{"Start":"01:27.500 ","End":"01:30.360","Text":"this and others who collected together and organized way,"},{"Start":"01:30.360 ","End":"01:33.110","Text":"and after that, you\u0027ll be equipped to answer"},{"Start":"01:33.110 ","End":"01:37.540","Text":"pretty much any reasonable homework question on the subject."},{"Start":"01:37.540 ","End":"01:41.530","Text":"That\u0027s in the following clips, and we\u0027re done here."}],"ID":26604},{"Watched":false,"Name":"Reduced Fractions","Duration":"2m 7s","ChapterTopicVideoID":25801,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this clip, we\u0027ll talk about reduced fractions concept you probably"},{"Start":"00:05.040 ","End":"00:09.840","Text":"are quite familiar with but reminder refresher is always good."},{"Start":"00:09.840 ","End":"00:13.620","Text":"Reduced fraction is 1 where the numerator and denominator have"},{"Start":"00:13.620 ","End":"00:19.995","Text":"no common factor that I know common factor except for 1 or minus 1."},{"Start":"00:19.995 ","End":"00:24.800","Text":"For example, 4/11 is reduced because there\u0027s no common factor for"},{"Start":"00:24.800 ","End":"00:29.795","Text":"4 and 11 and 4/10 is not reduced the both divisible by 2,"},{"Start":"00:29.795 ","End":"00:32.570","Text":"so we can take 2 out to the top and the bottom,"},{"Start":"00:32.570 ","End":"00:35.765","Text":"have say top and bottom instead of numerator and denominator."},{"Start":"00:35.765 ","End":"00:40.540","Text":"Then this reduces to 2/5 which is now reduced."},{"Start":"00:40.540 ","End":"00:44.600","Text":"As an example, when we got from 4/10 to 2/5,"},{"Start":"00:44.600 ","End":"00:47.390","Text":"every fraction can be written as a reduced fraction."},{"Start":"00:47.390 ","End":"00:50.045","Text":"You just keep reducing it that you can\u0027t do it anymore."},{"Start":"00:50.045 ","End":"00:53.615","Text":"Reduced fraction, I was going to say is unique,"},{"Start":"00:53.615 ","End":"00:55.750","Text":"but almost, not quite,"},{"Start":"00:55.750 ","End":"00:58.760","Text":"except for a sign reversal top and bottom,"},{"Start":"00:58.760 ","End":"01:01.880","Text":"so 2/5 is minus 2/minus 5."},{"Start":"01:01.880 ","End":"01:04.235","Text":"This is also a reduced fraction,"},{"Start":"01:04.235 ","End":"01:09.375","Text":"and 4/minus 11 is the same as minus 4/11."},{"Start":"01:09.375 ","End":"01:10.700","Text":"If you want to make it unique,"},{"Start":"01:10.700 ","End":"01:14.630","Text":"we can require that the denominator be positive."},{"Start":"01:14.630 ","End":"01:18.575","Text":"I\u0027d say that this is the reduced fraction,"},{"Start":"01:18.575 ","End":"01:22.550","Text":"and this 1 is the reduced fraction."},{"Start":"01:22.550 ","End":"01:26.005","Text":"The denominator is positive."},{"Start":"01:26.005 ","End":"01:28.525","Text":"That\u0027s not really important."},{"Start":"01:28.525 ","End":"01:33.830","Text":"Now, there are equivalent ways of saying that a fraction is reduced."},{"Start":"01:33.830 ","End":"01:36.920","Text":"Note that the following 3 conditions are equivalent,"},{"Start":"01:36.920 ","End":"01:39.505","Text":"a over b is a reduced fraction,"},{"Start":"01:39.505 ","End":"01:42.015","Text":"a and b are relatively prime,"},{"Start":"01:42.015 ","End":"01:44.280","Text":"sometimes use the word co-prime,"},{"Start":"01:44.280 ","End":"01:47.760","Text":"and the third possibility is that the GCD,"},{"Start":"01:47.760 ","End":"01:51.615","Text":"the Greatest Common Divisor of a and b is 1."},{"Start":"01:51.615 ","End":"01:54.875","Text":"Like here, 4 and 11."},{"Start":"01:54.875 ","End":"01:56.760","Text":"The greatest common divisor is 1,"},{"Start":"01:56.760 ","End":"02:01.290","Text":"but the greatest common divisor of 4 and 10 is 2."},{"Start":"02:01.310 ","End":"02:08.520","Text":"That\u0027s really all I wanted to say for now about reduced fractions. We\u0027re done."}],"ID":26605},{"Watched":false,"Name":"Fundamental Theorem of Arithmetic","Duration":"5m 35s","ChapterTopicVideoID":25799,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:07.035","Text":"In this clip, we\u0027ll learn about the fundamental theorem of arithmetic, FTA for short."},{"Start":"00:07.035 ","End":"00:12.675","Text":"Now, the goal that we have is to prove that certain numbers are irrational,"},{"Start":"00:12.675 ","End":"00:18.240","Text":"but we\u0027re still missing a main tool and that\u0027s this fundamental theorem of arithmetic."},{"Start":"00:18.240 ","End":"00:21.315","Text":"Let me state it although we won\u0027t prove it."},{"Start":"00:21.315 ","End":"00:26.595","Text":"Every positive integer can be expressed as a product of prime numbers."},{"Start":"00:26.595 ","End":"00:28.080","Text":"Thus, not all this."},{"Start":"00:28.080 ","End":"00:32.895","Text":"Representation is unique, except for the order of the factors."},{"Start":"00:32.895 ","End":"00:39.570","Text":"For example, 1,200 is 2^4 times 3 times 5 squared."},{"Start":"00:39.570 ","End":"00:46.025","Text":"Or if we write it out in full 2 times 2 times 2 times 2 times 3 times 5 times 5."},{"Start":"00:46.025 ","End":"00:47.630","Text":"It\u0027s a product of primes."},{"Start":"00:47.630 ","End":"00:50.090","Text":"But we can also write it if we rearrange the order as"},{"Start":"00:50.090 ","End":"00:53.185","Text":"5 times 2 times 5 times 2 times 3 times 2 times 2,"},{"Start":"00:53.185 ","End":"00:56.330","Text":"still the same factors just in a different order,"},{"Start":"00:56.330 ","End":"00:58.595","Text":"so it\u0027s unique except for the order."},{"Start":"00:58.595 ","End":"01:02.975","Text":"Another example, 210 is 2 times 3 times 5 times 7,"},{"Start":"01:02.975 ","End":"01:05.690","Text":"but it\u0027s also 7 times 2 times 5 times 3."},{"Start":"01:05.690 ","End":"01:08.630","Text":"It\u0027s the same factors just in a different order."},{"Start":"01:08.630 ","End":"01:11.780","Text":"You can\u0027t get something new except for rearranging the order."},{"Start":"01:11.780 ","End":"01:14.570","Text":"Another example, 945,"},{"Start":"01:14.570 ","End":"01:17.180","Text":"3 cubed times 5 times 7."},{"Start":"01:17.180 ","End":"01:21.335","Text":"I could also write it as 3 times 7 times 3 times 5 times 3."},{"Start":"01:21.335 ","End":"01:24.410","Text":"The theorem has 2 things for this example."},{"Start":"01:24.410 ","End":"01:29.780","Text":"First, that 1,200 can be represented as a product of primes."},{"Start":"01:29.780 ","End":"01:32.420","Text":"Secondly, that no matter how this is done,"},{"Start":"01:32.420 ","End":"01:35.030","Text":"there will always be exactly 4 2\u0027s,"},{"Start":"01:35.030 ","End":"01:39.500","Text":"1 3\u0027s, 2 5\u0027s and no other primes in the product."},{"Start":"01:39.500 ","End":"01:42.650","Text":"That\u0027s the fundamental theorem of arithmetic."},{"Start":"01:42.650 ","End":"01:44.405","Text":"Now let\u0027s continue."},{"Start":"01:44.405 ","End":"01:47.180","Text":"Is an application of the theorem."},{"Start":"01:47.180 ","End":"01:53.975","Text":"Suppose that p and a and b are natural numbers and p is a prime number."},{"Start":"01:53.975 ","End":"01:58.130","Text":"The claim is that if p divides a times b,"},{"Start":"01:58.130 ","End":"02:03.155","Text":"then p divides a or p divides b or both."},{"Start":"02:03.155 ","End":"02:05.725","Text":"Let\u0027s prove this."},{"Start":"02:05.725 ","End":"02:08.865","Text":"If p divides a times b,"},{"Start":"02:08.865 ","End":"02:12.420","Text":"then ab is p times something and we\u0027ll call that something,"},{"Start":"02:12.420 ","End":"02:18.590","Text":"c. So ab is pc for some natural number c. By the theorem,"},{"Start":"02:18.590 ","End":"02:21.540","Text":"we can write c equals p_1,"},{"Start":"02:21.540 ","End":"02:24.565","Text":"p_2 up to p_n."},{"Start":"02:24.565 ","End":"02:26.255","Text":"These are primes."},{"Start":"02:26.255 ","End":"02:28.165","Text":"There might be equal,"},{"Start":"02:28.165 ","End":"02:30.070","Text":"some of them, doesn\u0027t matter."},{"Start":"02:30.070 ","End":"02:36.255","Text":"Ab, which is p times c is p times p_1 through p_n."},{"Start":"02:36.255 ","End":"02:40.250","Text":"Now, a and b are also product of primes."},{"Start":"02:40.250 ","End":"02:42.050","Text":"To distinguish them,"},{"Start":"02:42.050 ","End":"02:45.070","Text":"I use dash prime,"},{"Start":"02:45.070 ","End":"02:46.710","Text":"I don\u0027t like these word prime,"},{"Start":"02:46.710 ","End":"02:50.010","Text":"it\u0027s ambiguous here, so let\u0027s say p\u0027 and p\"."},{"Start":"02:50.010 ","End":"02:52.470","Text":"P_i\u0027 and p_j\","},{"Start":"02:52.470 ","End":"02:58.390","Text":"i goes from 1 to k and j goes from 1 to m. These are all prime numbers."},{"Start":"02:58.390 ","End":"03:04.490","Text":"Pc is equal to p times this expansion for c and"},{"Start":"03:04.490 ","End":"03:10.655","Text":"this will be the expansion for a and this expansion product of primes for b."},{"Start":"03:10.655 ","End":"03:14.540","Text":"Now we\u0027ll use the uniqueness part of the fundamental theorem."},{"Start":"03:14.540 ","End":"03:18.235","Text":"The representation is unique except for the order."},{"Start":"03:18.235 ","End":"03:21.619","Text":"If it\u0027s unique, there\u0027s a p on the left."},{"Start":"03:21.619 ","End":"03:24.035","Text":"So there has to be a p on the right."},{"Start":"03:24.035 ","End":"03:25.460","Text":"The order is not important,"},{"Start":"03:25.460 ","End":"03:30.750","Text":"but 1 of these least must be p. Either 1 of the"},{"Start":"03:30.750 ","End":"03:38.030","Text":"p_i\u0027 or the p_j\" is equal to p. In the first case where p is 1 of these,"},{"Start":"03:38.030 ","End":"03:42.485","Text":"then p divides a and in the second case where p\u0027s 1 of these,"},{"Start":"03:42.485 ","End":"03:44.000","Text":"then p divides b,"},{"Start":"03:44.000 ","End":"03:45.750","Text":"but it could divide both."},{"Start":"03:45.750 ","End":"03:50.270","Text":"QED, that\u0027s what we have to prove for this claim."},{"Start":"03:50.270 ","End":"03:53.495","Text":"Now we can actually extend this claim,"},{"Start":"03:53.495 ","End":"03:57.125","Text":"doesn\u0027t have to be just 2 numbers, a and b."},{"Start":"03:57.125 ","End":"03:58.970","Text":"We could do it for 3 numbers."},{"Start":"03:58.970 ","End":"04:00.650","Text":"If p divides a, b, c,"},{"Start":"04:00.650 ","End":"04:03.710","Text":"then p divides a or p divides b or p divides"},{"Start":"04:03.710 ","End":"04:10.610","Text":"c. Because we can always break this up into ab times c. So p divides ab,"},{"Start":"04:10.610 ","End":"04:15.790","Text":"or p divides c. P divides ab means p divides a or p divides b."},{"Start":"04:15.790 ","End":"04:18.430","Text":"But we can generalize to more than 3."},{"Start":"04:18.430 ","End":"04:21.250","Text":"If p divides a_1, a_2 times,"},{"Start":"04:21.250 ","End":"04:22.875","Text":"and so on up to a_n,"},{"Start":"04:22.875 ","End":"04:25.030","Text":"then p divides 1 of these, a\u0027s,"},{"Start":"04:25.030 ","End":"04:28.280","Text":"1 of the a_i for at least 1 of them."},{"Start":"04:28.280 ","End":"04:34.535","Text":"The proof, well, it\u0027s just follows by induction it\u0027s like I said for this."},{"Start":"04:34.535 ","End":"04:39.110","Text":"You could say that if we take a_1 to a_n plus 1,"},{"Start":"04:39.110 ","End":"04:41.320","Text":"then p divides a_n plus 1,"},{"Start":"04:41.320 ","End":"04:46.920","Text":"or p divides 1 of these and then that breaks up into p divides 1 of the a_i."},{"Start":"04:46.920 ","End":"04:52.105","Text":"It\u0027s not that important to prove this should be clear if not, never mind."},{"Start":"04:52.105 ","End":"04:55.885","Text":"Now corollary, if p divides a^n,"},{"Start":"04:55.885 ","End":"04:57.190","Text":"where a is some integer,"},{"Start":"04:57.190 ","End":"05:00.705","Text":"then p divides a and p is a prime number."},{"Start":"05:00.705 ","End":"05:05.590","Text":"The proof of this is just that all the a_i\u0027s be equal to the same a."},{"Start":"05:05.590 ","End":"05:08.230","Text":"If p divides a times a times a times a,"},{"Start":"05:08.230 ","End":"05:11.935","Text":"then p divides a or p divides a or p divides a,"},{"Start":"05:11.935 ","End":"05:14.580","Text":"or p divides a. Am I right?"},{"Start":"05:14.580 ","End":"05:15.870","Text":"Or am I right?"},{"Start":"05:15.870 ","End":"05:17.550","Text":"I think you get the idea."},{"Start":"05:17.550 ","End":"05:19.340","Text":"It follows, for example,"},{"Start":"05:19.340 ","End":"05:21.230","Text":"if you take p equals 2,"},{"Start":"05:21.230 ","End":"05:23.600","Text":"that if a^n is an even number,"},{"Start":"05:23.600 ","End":"05:31.200","Text":"then so is a and this will be useful when we prove that the nth root of 2 is irrational."},{"Start":"05:31.330 ","End":"05:36.570","Text":"Come to think of it. That\u0027s all I want to say for this clip so we\u0027re done."}],"ID":26603},{"Watched":false,"Name":"Exercise 1","Duration":"3m 30s","ChapterTopicVideoID":8222,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we\u0027re going to prove that root 2 is irrational."},{"Start":"00:04.500 ","End":"00:05.760","Text":"That\u0027s part B."},{"Start":"00:05.760 ","End":"00:07.305","Text":"There\u0027s also a part A,"},{"Start":"00:07.305 ","End":"00:10.680","Text":"where we have to show that if a squared is even,"},{"Start":"00:10.680 ","End":"00:12.615","Text":"then a is even."},{"Start":"00:12.615 ","End":"00:14.685","Text":"Let\u0027s start with part a."},{"Start":"00:14.685 ","End":"00:19.430","Text":"We proved earlier that if a prime number p divides a to the n,"},{"Start":"00:19.430 ","End":"00:22.625","Text":"then p also divides a."},{"Start":"00:22.625 ","End":"00:25.485","Text":"In particular, take p equals 2."},{"Start":"00:25.485 ","End":"00:27.210","Text":"If 2 divides a to the n,"},{"Start":"00:27.210 ","End":"00:30.515","Text":"and 2 divides a if 2 is a prime number."},{"Start":"00:30.515 ","End":"00:33.410","Text":"Now if we let n equals to this n,"},{"Start":"00:33.410 ","End":"00:35.780","Text":"we get that if 2 divides a squared,"},{"Start":"00:35.780 ","End":"00:37.835","Text":"then 2 divides a."},{"Start":"00:37.835 ","End":"00:41.060","Text":"In other words, if a squared is even,"},{"Start":"00:41.060 ","End":"00:42.790","Text":"then a is even,"},{"Start":"00:42.790 ","End":"00:45.020","Text":"that\u0027s what we had to prove."},{"Start":"00:45.020 ","End":"00:46.805","Text":"This is so quick."},{"Start":"00:46.805 ","End":"00:49.055","Text":"Let me show you an alternative proof."},{"Start":"00:49.055 ","End":"00:52.355","Text":"What we\u0027ll do is we\u0027ll prove what\u0027s logically equivalent,"},{"Start":"00:52.355 ","End":"00:54.810","Text":"something called the counter positive."},{"Start":"00:54.810 ","End":"00:58.390","Text":"If statement p implies a statement q,"},{"Start":"00:58.390 ","End":"01:02.450","Text":"then not q implies not p, that\u0027s equivalent."},{"Start":"01:02.450 ","End":"01:06.280","Text":"Instead of proving, this will prove that if a is odd,"},{"Start":"01:06.280 ","End":"01:07.930","Text":"then a squared is odd."},{"Start":"01:07.930 ","End":"01:09.880","Text":"That will be logically equivalent."},{"Start":"01:09.880 ","End":"01:12.400","Text":"If a is odd, then a squared is odd,"},{"Start":"01:12.400 ","End":"01:15.130","Text":"intuitive that odd times odd is odd,"},{"Start":"01:15.130 ","End":"01:17.590","Text":"but let\u0027s actually prove it if a is odd,"},{"Start":"01:17.590 ","End":"01:19.330","Text":"then a can be written in the form 2,"},{"Start":"01:19.330 ","End":"01:22.015","Text":"n plus 1, or n is a whole number."},{"Start":"01:22.015 ","End":"01:25.315","Text":"That means that a squared is 2m plus 1 squared,"},{"Start":"01:25.315 ","End":"01:29.400","Text":"which if you expand this 4n square plus 4m plus 1 and we"},{"Start":"01:29.400 ","End":"01:33.420","Text":"can write this as twice 2n squared plus 2n plus 1,"},{"Start":"01:33.420 ","End":"01:35.490","Text":"which is twice something plus 1,"},{"Start":"01:35.490 ","End":"01:37.035","Text":"call this thing n,"},{"Start":"01:37.035 ","End":"01:38.855","Text":"which means that a squared,"},{"Start":"01:38.855 ","End":"01:41.570","Text":"which is this is also an odd number."},{"Start":"01:41.570 ","End":"01:44.540","Text":"That\u0027s the alternative proof of part A."},{"Start":"01:44.540 ","End":"01:45.860","Text":"Now let\u0027s get on to part B,"},{"Start":"01:45.860 ","End":"01:50.230","Text":"the main theorem that square root of 2 is irrational."},{"Start":"01:50.230 ","End":"01:52.965","Text":"We\u0027ll do this by contradiction,"},{"Start":"01:52.965 ","End":"01:59.000","Text":"so we\u0027ll suppose that root 2 is rational and then hopefully reach a contradiction."},{"Start":"01:59.000 ","End":"02:01.190","Text":"Root 2 is a over b,"},{"Start":"02:01.190 ","End":"02:03.610","Text":"where a and b are whole numbers."},{"Start":"02:03.610 ","End":"02:08.480","Text":"We can also assume that a over b is already reduced."},{"Start":"02:08.480 ","End":"02:13.490","Text":"It isn\u0027t reduced, then just keep producing it until we get irreducible,"},{"Start":"02:13.490 ","End":"02:19.895","Text":"and this part\u0027s important to assume that a over b is a reduced fraction."},{"Start":"02:19.895 ","End":"02:24.940","Text":"In fact, that root 2 is a over b means that 2 is a over b squared,"},{"Start":"02:24.940 ","End":"02:27.125","Text":"which is a squared over b squared."},{"Start":"02:27.125 ","End":"02:30.900","Text":"This gives us that a squared is 2b squared,"},{"Start":"02:30.900 ","End":"02:34.675","Text":"that means that a squared is even, it\u0027s twice something."},{"Start":"02:34.675 ","End":"02:36.325","Text":"If a squared is even,"},{"Start":"02:36.325 ","End":"02:39.910","Text":"then by part a, we have that a is even."},{"Start":"02:39.910 ","End":"02:42.340","Text":"If a is even, you can write it as twice something,"},{"Start":"02:42.340 ","End":"02:47.011","Text":"say twice c. Now, copying this a squared equals to b squared"},{"Start":"02:47.011 ","End":"02:49.135","Text":"and adding a equals 2c,"},{"Start":"02:49.135 ","End":"02:52.235","Text":"we get the 2c squared is 2b squared."},{"Start":"02:52.235 ","End":"02:55.540","Text":"This is 4c squared and divide both sides by 2,"},{"Start":"02:55.540 ","End":"03:00.010","Text":"so we get 2c squared equals b squared and that"},{"Start":"03:00.010 ","End":"03:04.900","Text":"gives us that b squared is even and hence that p is even."},{"Start":"03:04.900 ","End":"03:08.285","Text":"Look, a is even and b is even."},{"Start":"03:08.285 ","End":"03:11.700","Text":"That means that 2 divides a, 2 divides b."},{"Start":"03:11.700 ","End":"03:14.300","Text":"The fraction a over b is not reduced."},{"Start":"03:14.300 ","End":"03:16.910","Text":"We can take a 2 out of numerator and denominator."},{"Start":"03:16.910 ","End":"03:20.480","Text":"We said that it is reduced and that\u0027s a contradiction."},{"Start":"03:20.480 ","End":"03:25.175","Text":"The contradiction came from assuming that root 2 is rational."},{"Start":"03:25.175 ","End":"03:27.815","Text":"That\u0027s wrong and so root 2 is irrational."},{"Start":"03:27.815 ","End":"03:30.000","Text":"QED."}],"ID":8375},{"Watched":false,"Name":"Exercise 2","Duration":"4m 45s","ChapterTopicVideoID":8223,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"The aim of this exercise is to prove that the square root of 3 is irrational,"},{"Start":"00:04.860 ","End":"00:06.555","Text":"but we do it in 2 stages."},{"Start":"00:06.555 ","End":"00:11.340","Text":"In part a, we have to show that if a squared is divisible by 3,"},{"Start":"00:11.340 ","End":"00:13.860","Text":"then a is also divisible by 3,"},{"Start":"00:13.860 ","End":"00:16.020","Text":"a is some whole number."},{"Start":"00:16.020 ","End":"00:18.210","Text":"We\u0027ll prove part a in 2 different ways."},{"Start":"00:18.210 ","End":"00:21.735","Text":"1 would be similar to the previous exercise,"},{"Start":"00:21.735 ","End":"00:26.235","Text":"and we base it on the general claim which we showed in the tutorial,"},{"Start":"00:26.235 ","End":"00:29.400","Text":"that if a prime number p divides a to the n,"},{"Start":"00:29.400 ","End":"00:32.355","Text":"then p also divides a."},{"Start":"00:32.355 ","End":"00:36.750","Text":"So let\u0027s use this and then substitute p equals 3,"},{"Start":"00:36.750 ","End":"00:37.994","Text":"which is a prime,"},{"Start":"00:37.994 ","End":"00:39.870","Text":"and n equals 2."},{"Start":"00:39.870 ","End":"00:43.980","Text":"We get that if 3 divides a squared,"},{"Start":"00:43.980 ","End":"00:45.885","Text":"then 3 divides a,"},{"Start":"00:45.885 ","End":"00:47.805","Text":"and that\u0027s what we have to prove."},{"Start":"00:47.805 ","End":"00:50.040","Text":"This means a squared is divisible by 3,"},{"Start":"00:50.040 ","End":"00:52.035","Text":"and this means a is divisible by 3."},{"Start":"00:52.035 ","End":"00:54.920","Text":"Let\u0027s do it another way from basics,"},{"Start":"00:54.920 ","End":"00:57.169","Text":"without using this result."},{"Start":"00:57.169 ","End":"01:00.920","Text":"Now, I\u0027ll use the logical counter positive,"},{"Start":"01:00.920 ","End":"01:06.650","Text":"which in general says that p implies q is equivalent"},{"Start":"01:06.650 ","End":"01:13.745","Text":"to not q implies not p. So this is equivalent to if a is not divisible by 3,"},{"Start":"01:13.745 ","End":"01:17.085","Text":"then a squared is not divisible by 3."},{"Start":"01:17.085 ","End":"01:22.680","Text":"Now the remainder of dividing any number by 3 is going to be 0,"},{"Start":"01:22.680 ","End":"01:23.895","Text":"1, or 2."},{"Start":"01:23.895 ","End":"01:25.580","Text":"If the remainder is anymore than 2,"},{"Start":"01:25.580 ","End":"01:27.785","Text":"then the quotient increases by 1."},{"Start":"01:27.785 ","End":"01:29.390","Text":"So 0, 1, or, 2."},{"Start":"01:29.390 ","End":"01:32.620","Text":"If it\u0027s 0, it\u0027s divisible by 3."},{"Start":"01:32.620 ","End":"01:34.515","Text":"If it\u0027s 1 or 2,"},{"Start":"01:34.515 ","End":"01:36.525","Text":"it\u0027s not divisible by 3,"},{"Start":"01:36.525 ","End":"01:38.310","Text":"which if and only if."},{"Start":"01:38.310 ","End":"01:42.590","Text":"Yeah, a is not divisible by 3 if and only if the remainder is 1 or 2."},{"Start":"01:42.590 ","End":"01:47.205","Text":"So either a is of the form 3k plus 1,"},{"Start":"01:47.205 ","End":"01:48.900","Text":"that\u0027s when the remainder is 1,"},{"Start":"01:48.900 ","End":"01:51.920","Text":"or a is 3k plus 2 when the remainder is 2,"},{"Start":"01:51.920 ","End":"01:53.375","Text":"where k is a whole number."},{"Start":"01:53.375 ","End":"01:56.225","Text":"What we\u0027ll do is we\u0027ll find the remainder for a squared."},{"Start":"01:56.225 ","End":"01:58.575","Text":"The remainder when you divide it by 3, I mean."},{"Start":"01:58.575 ","End":"02:00.540","Text":"In each of the 2 cases."},{"Start":"02:00.540 ","End":"02:02.020","Text":"First of all, this case,"},{"Start":"02:02.020 ","End":"02:03.500","Text":"a equals 3k plus 1."},{"Start":"02:03.500 ","End":"02:07.900","Text":"Square it, a squared is 9k squared plus 6k plus 1."},{"Start":"02:07.900 ","End":"02:11.930","Text":"Can organize this as 3 times something plus 1,"},{"Start":"02:11.930 ","End":"02:13.520","Text":"so the remainder is 1."},{"Start":"02:13.520 ","End":"02:14.690","Text":"In the other case,"},{"Start":"02:14.690 ","End":"02:16.865","Text":"when a is 3k plus 2,"},{"Start":"02:16.865 ","End":"02:18.835","Text":"we get this expression."},{"Start":"02:18.835 ","End":"02:22.295","Text":"Again we can write it as 3 times something plus 1."},{"Start":"02:22.295 ","End":"02:25.160","Text":"So in either case, the remainder is 1."},{"Start":"02:25.160 ","End":"02:29.525","Text":"The remainder of 1 means that it\u0027s not divisible by 3."},{"Start":"02:29.525 ","End":"02:32.840","Text":"That\u0027s what we had to prove for part a."},{"Start":"02:32.840 ","End":"02:35.555","Text":"So now let\u0027s move on to part b."},{"Start":"02:35.555 ","End":"02:38.525","Text":"Part b, we\u0027ll do by contradiction."},{"Start":"02:38.525 ","End":"02:42.485","Text":"Remember we have to prove that the square root of 3 is irrational."},{"Start":"02:42.485 ","End":"02:49.175","Text":"Suppose that root 3 is rational and we\u0027ll reach a contradiction hopefully."},{"Start":"02:49.175 ","End":"02:54.785","Text":"So root 3 is of the form a over b because that\u0027s what a rational number is,"},{"Start":"02:54.785 ","End":"02:56.710","Text":"a and b are whole numbers."},{"Start":"02:56.710 ","End":"03:01.160","Text":"We can also assume that a over b is a reduced fraction."},{"Start":"03:01.160 ","End":"03:04.310","Text":"If it\u0027s not reduced, then reduce it."},{"Start":"03:04.310 ","End":"03:06.050","Text":"Every fraction can be reduced,"},{"Start":"03:06.050 ","End":"03:08.210","Text":"just simply divide top and bottom by"},{"Start":"03:08.210 ","End":"03:12.320","Text":"any common divisor and keep going until you can\u0027t do it anymore."},{"Start":"03:12.320 ","End":"03:15.070","Text":"Now if a over b is root 3,"},{"Start":"03:15.070 ","End":"03:17.205","Text":"by definition of square root,"},{"Start":"03:17.205 ","End":"03:19.705","Text":"a over b squared is 3."},{"Start":"03:19.705 ","End":"03:22.610","Text":"Then by laws of exponents,"},{"Start":"03:22.610 ","End":"03:25.220","Text":"this is equal to a squared over b squared."},{"Start":"03:25.220 ","End":"03:30.655","Text":"Multiply both sides by b squared and we get that a squared equals 3 b squared."},{"Start":"03:30.655 ","End":"03:36.155","Text":"That means that a squared is divisible by 3 because it\u0027s 3 times a whole number."},{"Start":"03:36.155 ","End":"03:39.109","Text":"Then by Part a,"},{"Start":"03:39.109 ","End":"03:42.140","Text":"we get that a is divisible by 3."},{"Start":"03:42.140 ","End":"03:43.640","Text":"The reason I\u0027ve colored this."},{"Start":"03:43.640 ","End":"03:45.230","Text":"If a is divisible by 3,"},{"Start":"03:45.230 ","End":"03:46.640","Text":"a is 3 times something,"},{"Start":"03:46.640 ","End":"03:51.800","Text":"let\u0027s say it\u0027s 3 times c. Now we can take this here,"},{"Start":"03:51.800 ","End":"03:55.040","Text":"copy it, and now replace a by 3c."},{"Start":"03:55.040 ","End":"03:58.210","Text":"So we\u0027ve got 3c squared is 3b squared."},{"Start":"03:58.210 ","End":"04:01.230","Text":"9c squared is 3b squared, divided by 3."},{"Start":"04:01.230 ","End":"04:04.035","Text":"3c squared equals b squared."},{"Start":"04:04.035 ","End":"04:07.955","Text":"Now we see that b squared is divisible by 3."},{"Start":"04:07.955 ","End":"04:10.820","Text":"Again, using Part a,"},{"Start":"04:10.820 ","End":"04:14.090","Text":"this implies that b itself is divisible by 3."},{"Start":"04:14.090 ","End":"04:19.280","Text":"So look, a is divisible by 3 and b is divisible by 3,"},{"Start":"04:19.280 ","End":"04:22.760","Text":"which means that a and b are both divisible by 3,"},{"Start":"04:22.760 ","End":"04:25.550","Text":"which means that they have a common factor."},{"Start":"04:25.550 ","End":"04:28.010","Text":"So it\u0027s not a reduced fraction."},{"Start":"04:28.010 ","End":"04:30.275","Text":"If it\u0027s not a reduced fraction,"},{"Start":"04:30.275 ","End":"04:32.860","Text":"that\u0027s a contradiction because we know it is."},{"Start":"04:32.860 ","End":"04:35.270","Text":"Where did the contradiction come from?"},{"Start":"04:35.270 ","End":"04:39.680","Text":"The contradiction came from supposing that square root of 3 is rational."},{"Start":"04:39.680 ","End":"04:42.370","Text":"So that can\u0027t be right, so it\u0027s irrational."},{"Start":"04:42.370 ","End":"04:45.780","Text":"That\u0027s what we had to show. So we\u0027re done."}],"ID":8376},{"Watched":false,"Name":"Exercise 3","Duration":"3m 35s","ChapterTopicVideoID":8224,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this exercise, we\u0027ll prove that cube root of"},{"Start":"00:03.960 ","End":"00:07.500","Text":"2 is an irrational number, but that\u0027s part B."},{"Start":"00:07.500 ","End":"00:09.630","Text":"Part A, we do some preparation for it."},{"Start":"00:09.630 ","End":"00:13.065","Text":"The claim is that if a cubed is even,"},{"Start":"00:13.065 ","End":"00:15.090","Text":"then a is even."},{"Start":"00:15.090 ","End":"00:18.420","Text":"We\u0027ll do this part A in 2 different ways."},{"Start":"00:18.420 ","End":"00:23.820","Text":"One, we\u0027ll use a little theorem proposition claim from earlier"},{"Start":"00:23.820 ","End":"00:29.505","Text":"that if a prime number p divides a^n, then p also divides a,"},{"Start":"00:29.505 ","End":"00:31.070","Text":"a is a whole number."},{"Start":"00:31.070 ","End":"00:36.980","Text":"If we just replace n by 3 and p by 2,"},{"Start":"00:36.980 ","End":"00:39.275","Text":"then we get exactly what we want."},{"Start":"00:39.275 ","End":"00:42.120","Text":"We get that if 2 divides a cubed,"},{"Start":"00:42.120 ","End":"00:43.325","Text":"then 2 divides a,"},{"Start":"00:43.325 ","End":"00:46.685","Text":"which in other words, means that if a cubed is even,"},{"Start":"00:46.685 ","End":"00:49.445","Text":"then a is even. That\u0027s it."},{"Start":"00:49.445 ","End":"00:51.170","Text":"Now, I said I\u0027ll prove it in 2 ways."},{"Start":"00:51.170 ","End":"00:54.030","Text":"One, without this fancy claim just from"},{"Start":"00:54.030 ","End":"00:59.960","Text":"basics to prove the logically equivalent claim that if a is odd,"},{"Start":"00:59.960 ","End":"01:02.155","Text":"then a cubed is odd."},{"Start":"01:02.155 ","End":"01:04.970","Text":"That proves this because we prove it by contradiction."},{"Start":"01:04.970 ","End":"01:07.175","Text":"We\u0027ll say, let a cubed be even."},{"Start":"01:07.175 ","End":"01:10.370","Text":"Suppose that a is not even but odd."},{"Start":"01:10.370 ","End":"01:11.985","Text":"Then a cubed is odd,"},{"Start":"01:11.985 ","End":"01:13.295","Text":"and that\u0027s a contradiction,"},{"Start":"01:13.295 ","End":"01:14.820","Text":"so a must be even."},{"Start":"01:14.820 ","End":"01:19.340","Text":"We just have to prove this part that an odd number cubed is odd. Seems clear."},{"Start":"01:19.340 ","End":"01:20.750","Text":"The odd times odd is odd,"},{"Start":"01:20.750 ","End":"01:22.130","Text":"but we\u0027ll prove it."},{"Start":"01:22.130 ","End":"01:28.080","Text":"If it\u0027s odd, then a is 2n plus 1 where n is some whole number,"},{"Start":"01:28.080 ","End":"01:34.640","Text":"and then a cubed using binomial expansion comes out to be this expression."},{"Start":"01:34.640 ","End":"01:35.840","Text":"I\u0027ll leave you to check it."},{"Start":"01:35.840 ","End":"01:39.850","Text":"We can rearrange this as twice something plus 1."},{"Start":"01:39.850 ","End":"01:42.735","Text":"This is a whole number, 2n plus 1."},{"Start":"01:42.735 ","End":"01:45.170","Text":"To summarize that, if a is odd,"},{"Start":"01:45.170 ","End":"01:47.060","Text":"then it\u0027s of the form 2n plus 1,"},{"Start":"01:47.060 ","End":"01:51.305","Text":"which means that a cubed is also of the form 2n plus 1 means a cubed is odd,"},{"Start":"01:51.305 ","End":"01:53.965","Text":"so a is odd then a cubed is odd."},{"Start":"01:53.965 ","End":"01:57.140","Text":"That\u0027s the alternative proof for part A."},{"Start":"01:57.140 ","End":"01:59.640","Text":"Now, let\u0027s get onto the important part B"},{"Start":"01:59.640 ","End":"02:02.435","Text":"to show that the cube root of 2 is an irrational number."},{"Start":"02:02.435 ","End":"02:04.055","Text":"Prove it by contradiction."},{"Start":"02:04.055 ","End":"02:06.830","Text":"Suppose that cube root of 2 is rational,"},{"Start":"02:06.830 ","End":"02:11.930","Text":"then cube root of 2 is a fraction of 2 whole numbers, a/b."},{"Start":"02:11.930 ","End":"02:16.970","Text":"We can assume that a/b is a reduced fraction because if it isn\u0027t reduced,"},{"Start":"02:16.970 ","End":"02:18.200","Text":"then just reduce it."},{"Start":"02:18.200 ","End":"02:22.755","Text":"Keep dividing by any common divisors until you can\u0027t do it anymore."},{"Start":"02:22.755 ","End":"02:27.870","Text":"We can assume that a/b is a reduced fraction, and that\u0027s important."},{"Start":"02:27.870 ","End":"02:31.100","Text":"This is the cube root of 2 that when you cube it, you get 2."},{"Start":"02:31.100 ","End":"02:32.900","Text":"That\u0027s definition of cube root."},{"Start":"02:32.900 ","End":"02:35.660","Text":"This is a cubed cubed over b cubed."},{"Start":"02:35.660 ","End":"02:40.145","Text":"We get that a cubed is twice b cubed."},{"Start":"02:40.145 ","End":"02:45.365","Text":"That means that a cubed is even, it\u0027s twice something."},{"Start":"02:45.365 ","End":"02:48.050","Text":"By part A,"},{"Start":"02:48.050 ","End":"02:49.790","Text":"we get that a is even,"},{"Start":"02:49.790 ","End":"02:54.280","Text":"and an even number can be written as twice something, say 2c."},{"Start":"02:54.280 ","End":"02:56.790","Text":"Back here, a cubed is 2b cubed."},{"Start":"02:56.790 ","End":"02:59.025","Text":"Now plug in a equals 2c,"},{"Start":"02:59.025 ","End":"03:03.765","Text":"we get to 2c cubed is 2b cubed or 8c cubed is to 2b cubed,"},{"Start":"03:03.765 ","End":"03:07.200","Text":"so b cubed is 4c cubed, and that\u0027s even."},{"Start":"03:07.200 ","End":"03:08.280","Text":"If you can\u0027t see it\u0027s even,"},{"Start":"03:08.280 ","End":"03:11.070","Text":"write it as twice 2c cubed, and that\u0027s 2 times something."},{"Start":"03:11.070 ","End":"03:14.595","Text":"It\u0027s even. B cubed is even,"},{"Start":"03:14.595 ","End":"03:16.215","Text":"and b is even."},{"Start":"03:16.215 ","End":"03:17.460","Text":"Look, a is even,"},{"Start":"03:17.460 ","End":"03:21.230","Text":"and b is even that means that a/b is not"},{"Start":"03:21.230 ","End":"03:24.020","Text":"a reduced fraction because both top and bottom are"},{"Start":"03:24.020 ","End":"03:27.005","Text":"divisible by 2, that\u0027s a contradiction."},{"Start":"03:27.005 ","End":"03:30.830","Text":"The contradiction came from supposing that cube root of 2 is rational,"},{"Start":"03:30.830 ","End":"03:35.370","Text":"that can\u0027t be, so it\u0027s irrational QED."}],"ID":8377},{"Watched":false,"Name":"Exercise 4","Duration":"3m 15s","ChapterTopicVideoID":8225,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"This exercise is about taking the square root of a whole number."},{"Start":"00:03.990 ","End":"00:05.475","Text":"I\u0027m going to paraphrase it."},{"Start":"00:05.475 ","End":"00:08.880","Text":"What it says is that there\u0027s only two possibilities."},{"Start":"00:08.880 ","End":"00:10.950","Text":"If you take the square root of a whole number,"},{"Start":"00:10.950 ","End":"00:15.345","Text":"it\u0027s either irrational or an integer."},{"Start":"00:15.345 ","End":"00:16.670","Text":"There\u0027s nothing in between."},{"Start":"00:16.670 ","End":"00:19.650","Text":"It\u0027s either very nice and it\u0027s a whole number,"},{"Start":"00:19.650 ","End":"00:21.360","Text":"or it completely irrational."},{"Start":"00:21.360 ","End":"00:23.415","Text":"I\u0027ll give example of one of each."},{"Start":"00:23.415 ","End":"00:26.610","Text":"Square root of 6 will be irrational,"},{"Start":"00:26.610 ","End":"00:29.235","Text":"and the square root of 9 will be an integer."},{"Start":"00:29.235 ","End":"00:33.270","Text":"You can\u0027t just get a rational number that\u0027s not a whole number."},{"Start":"00:33.270 ","End":"00:35.325","Text":"Let\u0027s prove it."},{"Start":"00:35.325 ","End":"00:39.510","Text":"Just an equivalent formulation is that,"},{"Start":"00:39.510 ","End":"00:42.590","Text":"if the square root of n is rational,"},{"Start":"00:42.590 ","End":"00:44.515","Text":"then it\u0027s an integer."},{"Start":"00:44.515 ","End":"00:47.165","Text":"Actually using some logic here,"},{"Start":"00:47.165 ","End":"00:53.695","Text":"if we say that p means that it\u0027s rational and q means that it\u0027s an integer,"},{"Start":"00:53.695 ","End":"00:57.550","Text":"then what we have here is p implies q."},{"Start":"00:57.550 ","End":"01:04.130","Text":"That\u0027s equivalent to the original formulation of not p or q,"},{"Start":"01:04.130 ","End":"01:09.310","Text":"p implies q is logically equivalent to not p or q."},{"Start":"01:09.310 ","End":"01:12.990","Text":"Let\u0027s say that square root of n is rational,"},{"Start":"01:12.990 ","End":"01:16.565","Text":"we have to conclude at the end that it\u0027s an integer,"},{"Start":"01:16.565 ","End":"01:19.070","Text":"root n as a over b,"},{"Start":"01:19.070 ","End":"01:22.010","Text":"where a over b is a reduced fraction."},{"Start":"01:22.010 ","End":"01:26.090","Text":"Any rational number can be written as a reduced fraction."},{"Start":"01:26.090 ","End":"01:28.000","Text":"If it\u0027s not reduced, then reduce it."},{"Start":"01:28.000 ","End":"01:30.350","Text":"We\u0027ll do a proof by contradiction."},{"Start":"01:30.350 ","End":"01:33.635","Text":"Suppose that root n is not an integer,"},{"Start":"01:33.635 ","End":"01:37.505","Text":"that means that b is not equal to 1,"},{"Start":"01:37.505 ","End":"01:39.895","Text":"because if it was an integer,"},{"Start":"01:39.895 ","End":"01:45.365","Text":"an integer as a reduced fraction has to have a denominator of 1 and vice versa."},{"Start":"01:45.365 ","End":"01:47.495","Text":"So b is not equal to 1,"},{"Start":"01:47.495 ","End":"01:52.355","Text":"so it has to be divisible by some number and hence by some prime number."},{"Start":"01:52.355 ","End":"01:54.980","Text":"You could use the fundamental theorem of arithmetic,"},{"Start":"01:54.980 ","End":"01:56.270","Text":"which is a bit of an overkill,"},{"Start":"01:56.270 ","End":"02:00.470","Text":"and say b is the product of primes and it\u0027s not an empty product."},{"Start":"02:00.470 ","End":"02:06.275","Text":"Choose one of them and that will be our p. Root n is a over b,"},{"Start":"02:06.275 ","End":"02:09.560","Text":"and that gives us that n is a over b squared,"},{"Start":"02:09.560 ","End":"02:11.815","Text":"which is a squared over b squared,"},{"Start":"02:11.815 ","End":"02:16.400","Text":"and that gives us that a squared is n times b squared."},{"Start":"02:16.400 ","End":"02:20.210","Text":"That means that p divides a squared,"},{"Start":"02:20.210 ","End":"02:21.650","Text":"because p divides b,"},{"Start":"02:21.650 ","End":"02:24.985","Text":"so p divides any product of b."},{"Start":"02:24.985 ","End":"02:27.500","Text":"If p divides a squared,"},{"Start":"02:27.500 ","End":"02:29.210","Text":"then p divides a."},{"Start":"02:29.210 ","End":"02:30.930","Text":"We\u0027ve seen this many times."},{"Start":"02:30.930 ","End":"02:32.900","Text":"In general, if p divides a to the n,"},{"Start":"02:32.900 ","End":"02:35.310","Text":"then p divides a."},{"Start":"02:36.680 ","End":"02:41.750","Text":"Now, p divides a and we already had p divides b,"},{"Start":"02:41.750 ","End":"02:47.120","Text":"so both a and b are divisible by p. If that\u0027s the case,"},{"Start":"02:47.120 ","End":"02:50.045","Text":"the fraction a over b is not reduced,"},{"Start":"02:50.045 ","End":"02:51.845","Text":"and that\u0027s a contradiction."},{"Start":"02:51.845 ","End":"02:57.665","Text":"The contradiction came from supposing that root n is not an integer."},{"Start":"02:57.665 ","End":"02:59.615","Text":"That supposition is false,"},{"Start":"02:59.615 ","End":"03:02.260","Text":"that means that root n is an integer."},{"Start":"03:02.260 ","End":"03:04.610","Text":"I think I\u0027ll summarize what we have so far."},{"Start":"03:04.610 ","End":"03:09.169","Text":"We showed that if the square root of n is a rational number,"},{"Start":"03:09.169 ","End":"03:10.835","Text":"then it\u0027s an integer,"},{"Start":"03:10.835 ","End":"03:15.600","Text":"and that\u0027s what we had to show QED and we\u0027re done."}],"ID":8378},{"Watched":false,"Name":"Exercise 5","Duration":"3m 45s","ChapterTopicVideoID":8226,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we have 5 parts."},{"Start":"00:02.850 ","End":"00:05.670","Text":"Each of them is a prove or disprove,"},{"Start":"00:05.670 ","End":"00:08.355","Text":"and when a statement is made,"},{"Start":"00:08.355 ","End":"00:09.600","Text":"like in the first one,"},{"Start":"00:09.600 ","End":"00:12.720","Text":"the product of 2 irrational numbers is irrational."},{"Start":"00:12.720 ","End":"00:15.360","Text":"That implies that it\u0027s always true."},{"Start":"00:15.360 ","End":"00:18.900","Text":"To disprove it, you would just need to find one example."},{"Start":"00:18.900 ","End":"00:20.760","Text":"Let\u0027s start with part A,"},{"Start":"00:20.760 ","End":"00:23.745","Text":"we\u0027ll just read each part as we come to it and then solve it."},{"Start":"00:23.745 ","End":"00:24.990","Text":"A is false,"},{"Start":"00:24.990 ","End":"00:26.865","Text":"and when I say false I mean,"},{"Start":"00:26.865 ","End":"00:28.695","Text":"that may be irrational,"},{"Start":"00:28.695 ","End":"00:30.630","Text":"but sometimes it isn\u0027t."},{"Start":"00:30.630 ","End":"00:32.385","Text":"Here\u0027s an example,"},{"Start":"00:32.385 ","End":"00:35.035","Text":"root 2 is irrational,"},{"Start":"00:35.035 ","End":"00:37.920","Text":"but if I multiply it by itself,"},{"Start":"00:37.920 ","End":"00:40.139","Text":"I\u0027ve got an irrational times an irrational,"},{"Start":"00:40.139 ","End":"00:42.255","Text":"but it gives me a rational."},{"Start":"00:42.255 ","End":"00:46.505","Text":"This is false in general but it could be true."},{"Start":"00:46.505 ","End":"00:49.190","Text":"For example, I could take root 2,"},{"Start":"00:49.190 ","End":"00:51.095","Text":"which is irrational times root 3,"},{"Start":"00:51.095 ","End":"00:53.120","Text":"which is irrational, and get root 6,"},{"Start":"00:53.120 ","End":"00:54.650","Text":"which is also irrational."},{"Start":"00:54.650 ","End":"00:56.270","Text":"It could be either,"},{"Start":"00:56.270 ","End":"00:58.915","Text":"as a general statement is false."},{"Start":"00:58.915 ","End":"01:02.720","Text":"Part B, the sum of 2 irrational numbers is irrational,"},{"Start":"01:02.720 ","End":"01:04.055","Text":"that\u0027s also false,"},{"Start":"01:04.055 ","End":"01:07.745","Text":"and a counterexample minus root 2 is irrational,"},{"Start":"01:07.745 ","End":"01:09.440","Text":"root 2 is irrational,"},{"Start":"01:09.440 ","End":"01:10.670","Text":"but if you add them,"},{"Start":"01:10.670 ","End":"01:13.715","Text":"you get 0, which is definitely rational."},{"Start":"01:13.715 ","End":"01:16.900","Text":"However, it could be irrational."},{"Start":"01:16.900 ","End":"01:21.430","Text":"You could add an irrational to an irrational root 2 plus root 2,"},{"Start":"01:21.430 ","End":"01:23.850","Text":"and it still comes out to be irrational,"},{"Start":"01:23.850 ","End":"01:26.295","Text":"it could be either in part C,"},{"Start":"01:26.295 ","End":"01:31.940","Text":"we want the quotient of 2 irrational numbers and is it always irrational here again,"},{"Start":"01:31.940 ","End":"01:34.685","Text":"it could be, it could not be."},{"Start":"01:34.685 ","End":"01:38.120","Text":"For example, root 2 is irrational,"},{"Start":"01:38.120 ","End":"01:42.855","Text":"but irrational over irrational root 2 over root 2 is 1 which is rational."},{"Start":"01:42.855 ","End":"01:47.360","Text":"Still, it could come out also irrational because like root 6 is irrational,"},{"Start":"01:47.360 ","End":"01:48.530","Text":"root 2 is irrational."},{"Start":"01:48.530 ","End":"01:52.110","Text":"If you divide them, it\u0027s still irrational so could be either,"},{"Start":"01:52.110 ","End":"01:57.000","Text":"but in general, the statement is false. What was part D?"},{"Start":"01:57.000 ","End":"02:03.690","Text":"The sum of 2 rational numbers is rational, that\u0027s definitely true."},{"Start":"02:03.690 ","End":"02:08.870","Text":"The proof, I have a rational number a over b and another rational number c over"},{"Start":"02:08.870 ","End":"02:14.210","Text":"d. We can actually compute it like so and if all these are integers,"},{"Start":"02:14.210 ","End":"02:16.460","Text":"then ad plus bc will be an integer,"},{"Start":"02:16.460 ","End":"02:18.305","Text":"and bd will be an integer."},{"Start":"02:18.305 ","End":"02:22.040","Text":"Rational plus rational is rational, that\u0027s always."},{"Start":"02:22.040 ","End":"02:29.210","Text":"The last one says the sum of the rational number and an irrational number is irrational."},{"Start":"02:29.210 ","End":"02:31.100","Text":"I claim that\u0027s true."},{"Start":"02:31.100 ","End":"02:36.035","Text":"Suppose that q is rational and p is irrational,"},{"Start":"02:36.035 ","End":"02:38.885","Text":"q belongs to q and p doesn\u0027t belong to q."},{"Start":"02:38.885 ","End":"02:44.195","Text":"We want to know if x is irrational and I\u0027ll prove it by contradiction,"},{"Start":"02:44.195 ","End":"02:46.855","Text":"that indeed x is irrational."},{"Start":"02:46.855 ","End":"02:51.380","Text":"By contradiction, suppose that this is false,"},{"Start":"02:51.380 ","End":"02:55.535","Text":"which means that it does belong to the rational numbers that x is rational."},{"Start":"02:55.535 ","End":"03:00.140","Text":"Now q is rational so also minus q is rational."},{"Start":"03:00.140 ","End":"03:05.600","Text":"Minus irrational number is also rational because the minus of a over b minus a over b,"},{"Start":"03:05.600 ","End":"03:08.075","Text":"which is also an integer over an integer."},{"Start":"03:08.075 ","End":"03:13.250","Text":"You can write p from here as minus q plus x."},{"Start":"03:13.250 ","End":"03:18.920","Text":"Now, q is rational, so minus q is rational and x is rational by"},{"Start":"03:18.920 ","End":"03:26.105","Text":"supposition and the sum of rational plus rational is rational by part D. However,"},{"Start":"03:26.105 ","End":"03:31.670","Text":"p is not rational by what we\u0027re given, that\u0027s a contradiction."},{"Start":"03:31.670 ","End":"03:33.440","Text":"On the one hand, it should be rational,"},{"Start":"03:33.440 ","End":"03:35.690","Text":"on the one hand, it isn\u0027t rational."},{"Start":"03:35.690 ","End":"03:40.834","Text":"That contradiction came from supposing that x is rational,"},{"Start":"03:40.834 ","End":"03:42.515","Text":"and so it\u0027s irrational."},{"Start":"03:42.515 ","End":"03:45.780","Text":"That proves what we wanted and we\u0027re done."}],"ID":8379},{"Watched":false,"Name":"Exercise 6","Duration":"5m 30s","ChapterTopicVideoID":8227,"CourseChapterTopicPlaylistID":200,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"In this exercise, we have 3 expressions,"},{"Start":"00:04.515 ","End":"00:08.130","Text":"and each of them is an irrational number,"},{"Start":"00:08.130 ","End":"00:11.620","Text":"or at least we\u0027ll know that after we\u0027ve proved this, and that\u0027s your task,"},{"Start":"00:11.620 ","End":"00:14.105","Text":"to prove that each of these is irrational,"},{"Start":"00:14.105 ","End":"00:16.500","Text":"and they get progressively more difficult."},{"Start":"00:16.500 ","End":"00:17.910","Text":"Let\u0027s start with the easy 1,"},{"Start":"00:17.910 ","End":"00:21.180","Text":"the first, and we\u0027ll do it by contradiction."},{"Start":"00:21.180 ","End":"00:25.230","Text":"We\u0027ll suppose that root 2 plus root 3 is rational,"},{"Start":"00:25.230 ","End":"00:27.150","Text":"we\u0027ll call it little q,"},{"Start":"00:27.150 ","End":"00:29.430","Text":"that\u0027s in the set of rational numbers."},{"Start":"00:29.430 ","End":"00:33.150","Text":"Square both sides, and then expand,"},{"Start":"00:33.150 ","End":"00:34.890","Text":"and we get this,"},{"Start":"00:34.890 ","End":"00:42.280","Text":"and we can extract root 6 from this and get that root 6 is q squared minus 5 over 2."},{"Start":"00:42.280 ","End":"00:44.840","Text":"There\u0027s an easy way to get to the answer from here."},{"Start":"00:44.840 ","End":"00:49.850","Text":"If you know that the rational numbers form a field,"},{"Start":"00:49.850 ","End":"00:52.220","Text":"you can add, subtract, multiply,"},{"Start":"00:52.220 ","End":"00:55.700","Text":"and divide and stay within the rational numbers."},{"Start":"00:55.700 ","End":"00:57.500","Text":"At least, if you know that the rational numbers are"},{"Start":"00:57.500 ","End":"00:59.660","Text":"closed under the 4 operations of addition,"},{"Start":"00:59.660 ","End":"01:01.795","Text":"subtraction, multiplication, division,"},{"Start":"01:01.795 ","End":"01:03.240","Text":"then q\u0027s rational,"},{"Start":"01:03.240 ","End":"01:04.470","Text":"so q squared is rational,"},{"Start":"01:04.470 ","End":"01:06.300","Text":"so minus 5 is still rational,"},{"Start":"01:06.300 ","End":"01:07.995","Text":"and divided by 2 is rational,"},{"Start":"01:07.995 ","End":"01:09.350","Text":"but I\u0027ll do it the long way,"},{"Start":"01:09.350 ","End":"01:10.730","Text":"assuming you don\u0027t know that."},{"Start":"01:10.730 ","End":"01:12.590","Text":"Let\u0027s say that q is a over b,"},{"Start":"01:12.590 ","End":"01:14.980","Text":"where a and b are whole numbers."},{"Start":"01:14.980 ","End":"01:17.790","Text":"Root 6, just plugging a over b,"},{"Start":"01:17.790 ","End":"01:19.695","Text":"and here is this."},{"Start":"01:19.695 ","End":"01:21.330","Text":"If you simplify this,"},{"Start":"01:21.330 ","End":"01:25.160","Text":"I won\u0027t go into the algebraic details, we get this,"},{"Start":"01:25.160 ","End":"01:27.620","Text":"which is a whole number over a whole number,"},{"Start":"01:27.620 ","End":"01:29.435","Text":"and therefore a rational number,"},{"Start":"01:29.435 ","End":"01:34.010","Text":"and that\u0027s a contradiction because we know that root 6 is not rational."},{"Start":"01:34.010 ","End":"01:38.089","Text":"The square root of a whole number is either a whole number or irrational,"},{"Start":"01:38.089 ","End":"01:40.000","Text":"and this is certainly not a whole number."},{"Start":"01:40.000 ","End":"01:42.540","Text":"Now, let\u0027s go on to part b,"},{"Start":"01:42.540 ","End":"01:48.705","Text":"which says that root 2 plus root 3 plus root 5 is irrational."},{"Start":"01:48.705 ","End":"01:50.445","Text":"Again, by contradiction,"},{"Start":"01:50.445 ","End":"01:53.220","Text":"we suppose that it\u0027s actually rational."},{"Start":"01:53.220 ","End":"01:55.820","Text":"We\u0027ll get a contradiction, and this will show that it\u0027s irrational."},{"Start":"01:55.820 ","End":"01:58.760","Text":"Bring the root 5 over to the other side."},{"Start":"01:58.760 ","End":"02:02.585","Text":"Next, square both sides and expand,"},{"Start":"02:02.585 ","End":"02:07.050","Text":"and here we have 2 plus twice root 2, root 3,"},{"Start":"02:07.050 ","End":"02:08.130","Text":"which is root 6,"},{"Start":"02:08.130 ","End":"02:09.870","Text":"and then root 3 squared,"},{"Start":"02:09.870 ","End":"02:12.540","Text":"which is 3, the 3 with the 2 give 5."},{"Start":"02:12.540 ","End":"02:14.130","Text":"Here we also have a 5."},{"Start":"02:14.130 ","End":"02:15.375","Text":"The 5 cancels."},{"Start":"02:15.375 ","End":"02:17.720","Text":"Anyway, square both sides."},{"Start":"02:17.720 ","End":"02:19.520","Text":"Here we have 2 root 6 squared,"},{"Start":"02:19.520 ","End":"02:22.800","Text":"and here we have q squared minus 2q root 5 squared,"},{"Start":"02:22.800 ","End":"02:25.610","Text":"but I took the q outside the brackets first,"},{"Start":"02:25.610 ","End":"02:29.210","Text":"so it\u0027s q times q minus 2 root 5 squared squared."},{"Start":"02:29.210 ","End":"02:31.550","Text":"This is 2 times 2 times 6,"},{"Start":"02:31.550 ","End":"02:32.995","Text":"which is 24,"},{"Start":"02:32.995 ","End":"02:36.830","Text":"equals q squared times this,"},{"Start":"02:36.830 ","End":"02:40.040","Text":"and then divide by q squared,"},{"Start":"02:40.040 ","End":"02:42.040","Text":"so that goes down here,"},{"Start":"02:42.040 ","End":"02:45.765","Text":"and then take the q squared and the 20,"},{"Start":"02:45.765 ","End":"02:47.370","Text":"and throw them over to the left,"},{"Start":"02:47.370 ","End":"02:49.650","Text":"so that\u0027s minus q squared minus 20,"},{"Start":"02:49.650 ","End":"02:51.180","Text":"and what we\u0027re left, on the right,"},{"Start":"02:51.180 ","End":"02:53.400","Text":"is minus 4q root 5."},{"Start":"02:53.400 ","End":"02:58.705","Text":"Now, we can extract root 5 by dividing both sides by minus 4q."},{"Start":"02:58.705 ","End":"03:03.139","Text":"Really, this shows that root 5 is a rational number."},{"Start":"03:03.139 ","End":"03:06.440","Text":"If you know that addition, multiplication, division,"},{"Start":"03:06.440 ","End":"03:09.350","Text":"subtraction of rational numbers is a rational number,"},{"Start":"03:09.350 ","End":"03:11.480","Text":"you could end here, but if not,"},{"Start":"03:11.480 ","End":"03:15.005","Text":"we can replace q by a over b,"},{"Start":"03:15.005 ","End":"03:17.105","Text":"plug a over b into here,"},{"Start":"03:17.105 ","End":"03:19.235","Text":"and after you simplify it,"},{"Start":"03:19.235 ","End":"03:20.895","Text":"this is what you get."},{"Start":"03:20.895 ","End":"03:23.510","Text":"Obviously, it\u0027s a whole number on top,"},{"Start":"03:23.510 ","End":"03:25.160","Text":"a whole number on the bottom,"},{"Start":"03:25.160 ","End":"03:28.505","Text":"which shows that root 5 is a rational number,"},{"Start":"03:28.505 ","End":"03:32.765","Text":"and that\u0027s the contradiction because we know that root 5 is not rational."},{"Start":"03:32.765 ","End":"03:36.740","Text":"The square root of a whole number is either a whole number or irrational,"},{"Start":"03:36.740 ","End":"03:38.405","Text":"and it\u0027s certainly not a whole number."},{"Start":"03:38.405 ","End":"03:41.440","Text":"That contradiction proves our case."},{"Start":"03:41.440 ","End":"03:44.190","Text":"Time to move on to part c. In part c,"},{"Start":"03:44.190 ","End":"03:48.535","Text":"we have to show that cube root of 2 plus square root of 3 is irrational,"},{"Start":"03:48.535 ","End":"03:50.450","Text":"and just like in the other cases,"},{"Start":"03:50.450 ","End":"03:52.100","Text":"were doing it by contradiction,"},{"Start":"03:52.100 ","End":"03:54.650","Text":"so supposing that it is a rational number."},{"Start":"03:54.650 ","End":"03:58.580","Text":"Now, bring the square root over to the right-hand side,"},{"Start":"03:58.580 ","End":"04:00.175","Text":"then switch sides,"},{"Start":"04:00.175 ","End":"04:04.790","Text":"so we have q minus root 3 equals cube root of 2."},{"Start":"04:04.790 ","End":"04:07.580","Text":"Now, raise both sides to the power of 3,"},{"Start":"04:07.580 ","End":"04:11.645","Text":"and what we\u0027ll use is the binomial formula,"},{"Start":"04:11.645 ","End":"04:14.540","Text":"x minus y cubed is the following."},{"Start":"04:14.540 ","End":"04:16.070","Text":"Here, x is q,"},{"Start":"04:16.070 ","End":"04:18.140","Text":"and y is root 3."},{"Start":"04:18.140 ","End":"04:20.690","Text":"What we get is the following."},{"Start":"04:20.690 ","End":"04:24.690","Text":"I\u0027m using the fact that root 3 times root 3 is 3,"},{"Start":"04:24.690 ","End":"04:28.460","Text":"so when I have y squared, like here,"},{"Start":"04:28.460 ","End":"04:30.305","Text":"that\u0027s the y squared, that\u0027s 3,"},{"Start":"04:30.305 ","End":"04:34.160","Text":"and y cubed is 3 root 3."},{"Start":"04:34.160 ","End":"04:35.865","Text":"Tidy this up a bit."},{"Start":"04:35.865 ","End":"04:38.985","Text":"Put the terms with root 3 on the right."},{"Start":"04:38.985 ","End":"04:42.635","Text":"Now, divide by what comes before the root 3,"},{"Start":"04:42.635 ","End":"04:46.065","Text":"and we have that root 3 is this expression,"},{"Start":"04:46.065 ","End":"04:47.460","Text":"q is rational,"},{"Start":"04:47.460 ","End":"04:50.930","Text":"so we could say right away that this is also rational because"},{"Start":"04:50.930 ","End":"04:56.055","Text":"some product difference and so on of rational numbers is rational,"},{"Start":"04:56.055 ","End":"05:00.410","Text":"or you could do it directly by replacing q with a over b,"},{"Start":"05:00.410 ","End":"05:02.120","Text":"where a over b whole numbers,"},{"Start":"05:02.120 ","End":"05:04.325","Text":"and after you do a little bit of work,"},{"Start":"05:04.325 ","End":"05:06.220","Text":"it simplifies to this."},{"Start":"05:06.220 ","End":"05:08.930","Text":"You can see that if a and b are whole numbers,"},{"Start":"05:08.930 ","End":"05:10.280","Text":"then the top is a whole number,"},{"Start":"05:10.280 ","End":"05:11.765","Text":"the bottom is a whole number,"},{"Start":"05:11.765 ","End":"05:15.350","Text":"so root 3 is a fraction of 2 whole numbers,"},{"Start":"05:15.350 ","End":"05:17.164","Text":"and so it\u0027s rational."},{"Start":"05:17.164 ","End":"05:21.020","Text":"That\u0027s a contradiction because we know that root 3 is not rational."},{"Start":"05:21.020 ","End":"05:27.015","Text":"That contradiction proves that that expression that we had is irrational,"},{"Start":"05:27.015 ","End":"05:30.850","Text":"and that concludes this exercise."}],"ID":8380}],"Thumbnail":null,"ID":200},{"Name":"Bounded and Unbounded Sets in R","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Bounded set","Duration":"3m 14s","ChapterTopicVideoID":25795,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"A new topic, Bounded and unbounded sets of real numbers."},{"Start":"00:05.280 ","End":"00:11.025","Text":"The first subject here is sets bounded from above and sets bounded from below."},{"Start":"00:11.025 ","End":"00:16.620","Text":"By the way, sets of numbers will all be sets of real numbers unless said otherwise."},{"Start":"00:16.620 ","End":"00:20.310","Text":"Our universal set is R, the real numbers."},{"Start":"00:20.310 ","End":"00:24.195","Text":"Now, as an example, consider the set A which is 2, 4, 6,"},{"Start":"00:24.195 ","End":"00:27.975","Text":"8, etc, the positive even numbers."},{"Start":"00:27.975 ","End":"00:34.125","Text":"I claim that there exists a number that\u0027s less than or equal to every member of this set,"},{"Start":"00:34.125 ","End":"00:35.940","Text":"exists more than 1 such."},{"Start":"00:35.940 ","End":"00:37.335","Text":"2 is an example."},{"Start":"00:37.335 ","End":"00:40.520","Text":"2 is less than or equal to everything in this set. But there are others."},{"Start":"00:40.520 ","End":"00:42.740","Text":"1 is also less than or equal to everything,"},{"Start":"00:42.740 ","End":"00:45.080","Text":"and minus 3 is less than or equal to everything,"},{"Start":"00:45.080 ","End":"00:46.550","Text":"and loads of others."},{"Start":"00:46.550 ","End":"00:50.195","Text":"In such a case that there\u0027s a number less than or equal to everything,"},{"Start":"00:50.195 ","End":"00:53.345","Text":"we say that the set is bounded from below,"},{"Start":"00:53.345 ","End":"00:55.535","Text":"so A is bounded from below."},{"Start":"00:55.535 ","End":"00:57.200","Text":"Each of these numbers,"},{"Start":"00:57.200 ","End":"00:59.210","Text":"2, 1, minus 3,"},{"Start":"00:59.210 ","End":"01:04.185","Text":"and loads of others are called a lower bound of A."},{"Start":"01:04.185 ","End":"01:08.825","Text":"Similarly, if we replace less than or equal to with greater than or equal to,"},{"Start":"01:08.825 ","End":"01:11.690","Text":"we could define a set that\u0027s bounded from above,"},{"Start":"01:11.690 ","End":"01:14.960","Text":"and we could define the concept of upper bound."},{"Start":"01:14.960 ","End":"01:18.760","Text":"Note that our set A has no upper bound."},{"Start":"01:18.760 ","End":"01:20.940","Text":"It grows to infinity."},{"Start":"01:20.940 ","End":"01:25.010","Text":"There\u0027s nothing that\u0027s bigger or equal to all the members of A,"},{"Start":"01:25.010 ","End":"01:27.260","Text":"so it\u0027s not bounded from above."},{"Start":"01:27.260 ","End":"01:30.105","Text":"But consider another example B,"},{"Start":"01:30.105 ","End":"01:31.500","Text":"which is 1, a 1/2, a 1/3/,"},{"Start":"01:31.500 ","End":"01:33.740","Text":"a 1/4, and so on."},{"Start":"01:33.740 ","End":"01:37.760","Text":"The set B is bounded from below."},{"Start":"01:37.760 ","End":"01:40.800","Text":"For example, 0 is a lower bound."},{"Start":"01:40.800 ","End":"01:42.150","Text":"This are all positive numbers,"},{"Start":"01:42.150 ","End":"01:44.104","Text":"so 0 is a lower bound."},{"Start":"01:44.104 ","End":"01:47.525","Text":"This time B is bounded from above."},{"Start":"01:47.525 ","End":"01:49.630","Text":"Well, I\u0027m asking the question first."},{"Start":"01:49.630 ","End":"01:54.185","Text":"Is there a number that\u0027s bigger or equal to every member of B?"},{"Start":"01:54.185 ","End":"01:55.580","Text":"Well, yes. 1,"},{"Start":"01:55.580 ","End":"01:57.935","Text":"for example, is bigger or equal to everything."},{"Start":"01:57.935 ","End":"02:00.440","Text":"But there\u0027s other examples like 4,"},{"Start":"02:00.440 ","End":"02:03.515","Text":"10 bigger or equal to every member of B."},{"Start":"02:03.515 ","End":"02:06.125","Text":"These are called upper bounds of B."},{"Start":"02:06.125 ","End":"02:11.420","Text":"Now another definition, if a set is bounded above and bounded below,"},{"Start":"02:11.420 ","End":"02:13.265","Text":"we should say bounded from above,"},{"Start":"02:13.265 ","End":"02:15.940","Text":"then we simply say that it is bounded."},{"Start":"02:15.940 ","End":"02:21.440","Text":"We saw that B is bounded because it\u0027s bounded above and below,"},{"Start":"02:21.440 ","End":"02:25.010","Text":"but A is not bounded because it\u0027s bounded from below,"},{"Start":"02:25.010 ","End":"02:28.639","Text":"but not bounded from above and needs to be both."},{"Start":"02:28.639 ","End":"02:32.285","Text":"Now let\u0027s give a more formal definition of a bounded set."},{"Start":"02:32.285 ","End":"02:35.795","Text":"Let\u0027s assume that A is a subset of the real numbers."},{"Start":"02:35.795 ","End":"02:40.955","Text":"A is said to be bounded from below if there exists a number m in"},{"Start":"02:40.955 ","End":"02:46.660","Text":"R such that m is less than or equal to x for all x in A."},{"Start":"02:46.660 ","End":"02:50.135","Text":"Such an m is called a lower bound of A."},{"Start":"02:50.135 ","End":"02:55.535","Text":"Similarly, A is said to be bounded from above if there exists an m in"},{"Start":"02:55.535 ","End":"03:01.085","Text":"R such that x is less than or equal to m for all x in A."},{"Start":"03:01.085 ","End":"03:04.280","Text":"Such an m is called an upper bound of A."},{"Start":"03:04.280 ","End":"03:08.585","Text":"Thirdly, A is said to be bounded,"},{"Start":"03:08.585 ","End":"03:10.070","Text":"just simply bounded,"},{"Start":"03:10.070 ","End":"03:14.489","Text":"if it is bounded from above and bounded from below."}],"ID":26599},{"Watched":false,"Name":"Supremum and Infimum","Duration":"5m 23s","ChapterTopicVideoID":25794,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:03.690","Text":"Continuing with bounded sets of real numbers"},{"Start":"00:03.690 ","End":"00:06.225","Text":"or some related concepts,"},{"Start":"00:06.225 ","End":"00:10.140","Text":"infimum, supremum, minimum and maximum."},{"Start":"00:10.140 ","End":"00:13.340","Text":"Let\u0027s just start with an example of the set"},{"Start":"00:13.340 ","End":"00:16.580","Text":"that we had in the previous clip are familiar set A,"},{"Start":"00:16.580 ","End":"00:19.390","Text":"which is the positive even numbers."},{"Start":"00:19.390 ","End":"00:22.910","Text":"As we saw, it has several lower bounds."},{"Start":"00:22.910 ","End":"00:24.440","Text":"We noted minus 4,"},{"Start":"00:24.440 ","End":"00:27.935","Text":"0, 2 actually has infinitely many."},{"Start":"00:27.935 ","End":"00:31.455","Text":"Now, I\u0027ve colored 1 of them because I think it\u0027s a special one,"},{"Start":"00:31.455 ","End":"00:34.160","Text":"there is 1 distinguished lower bound"},{"Start":"00:34.160 ","End":"00:35.470","Text":"and that\u0027s the 2."},{"Start":"00:35.470 ","End":"00:38.615","Text":"The reason is, is that it\u0027s the largest."},{"Start":"00:38.615 ","End":"00:44.110","Text":"Any other lower bound for this set has to be less than 2."},{"Start":"00:44.110 ","End":"00:49.115","Text":"Rephrase that, there\u0027s no lower bound of a greater than 2."},{"Start":"00:49.115 ","End":"00:52.190","Text":"In anything greater than 2 won\u0027t be a lower bound."},{"Start":"00:52.190 ","End":"00:54.085","Text":"It can\u0027t bound 2."},{"Start":"00:54.085 ","End":"00:59.780","Text":"Such a lower bound in general for a set is called the infimum."},{"Start":"00:59.780 ","End":"01:01.400","Text":"It\u0027s a lower bound,"},{"Start":"01:01.400 ","End":"01:04.270","Text":"which is the greatest among all lower bounds."},{"Start":"01:04.270 ","End":"01:08.710","Text":"It\u0027s called the infimum and notation I-N-F inf."},{"Start":"01:08.710 ","End":"01:12.200","Text":"Inf A is equal to 2."},{"Start":"01:12.200 ","End":"01:16.115","Text":"Infimum is also called the greatest lower bound,"},{"Start":"01:16.115 ","End":"01:20.540","Text":"abbreviated g.l.b of A obvious reasons"},{"Start":"01:20.540 ","End":"01:21.710","Text":"because it\u0027s a lower bound"},{"Start":"01:21.710 ","End":"01:24.215","Text":"and it\u0027s the greatest among all lower bounds."},{"Start":"01:24.215 ","End":"01:27.260","Text":"Now recall our set B from earlier,"},{"Start":"01:27.260 ","End":"01:28.550","Text":"1, a half, a third,"},{"Start":"01:28.550 ","End":"01:31.250","Text":"a quarter, 1 over n in general,"},{"Start":"01:31.250 ","End":"01:34.090","Text":"where n is a positive natural number."},{"Start":"01:34.090 ","End":"01:40.354","Text":"We already said that it has a lower bound and that lower bound is 0."},{"Start":"01:40.354 ","End":"01:44.060","Text":"I claim that 0 is the greatest lower bound i.e."},{"Start":"01:44.060 ","End":"01:47.135","Text":"the infimum. But intuitively,"},{"Start":"01:47.135 ","End":"01:50.870","Text":"we can\u0027t have positive lower bound."},{"Start":"01:50.870 ","End":"01:54.710","Text":"1 over n will eventually be smaller than any positive number."},{"Start":"01:54.710 ","End":"01:58.990","Text":"It can\u0027t be a lower bound for all the elements."},{"Start":"01:58.990 ","End":"02:06.245","Text":"0 is a lower bound and there\u0027s nothing greater for which the greatest lower bound,"},{"Start":"02:06.245 ","End":"02:08.480","Text":"and that makes it the infimum."},{"Start":"02:08.480 ","End":"02:12.110","Text":"Note that 0 itself is not in the set."},{"Start":"02:12.110 ","End":"02:13.525","Text":"All the elements here,"},{"Start":"02:13.525 ","End":"02:17.430","Text":"are 1 over n. They\u0027re all positive."},{"Start":"02:17.430 ","End":"02:19.169","Text":"It\u0027s the infimum."},{"Start":"02:19.169 ","End":"02:22.595","Text":"But note that it doesn\u0027t belong to the set B."},{"Start":"02:22.595 ","End":"02:25.650","Text":"This is a contrast between A and B."},{"Start":"02:25.650 ","End":"02:27.255","Text":"In the case of A,"},{"Start":"02:27.255 ","End":"02:30.090","Text":"the infimum is 2."},{"Start":"02:30.090 ","End":"02:32.235","Text":"It does belong to A."},{"Start":"02:32.235 ","End":"02:34.090","Text":"In the case of B the infimum,"},{"Start":"02:34.090 ","End":"02:37.615","Text":"which is 0, does not belong to B."},{"Start":"02:37.615 ","End":"02:41.280","Text":"If the infimum happens to belong to the set"},{"Start":"02:41.280 ","End":"02:45.430","Text":"and it\u0027s also called the minimum of the set."},{"Start":"02:45.430 ","End":"02:48.900","Text":"So that 2 is the minimum of A"},{"Start":"02:48.900 ","End":"02:54.775","Text":"and we write min A equals 2 but 0 is not the minimum of B."},{"Start":"02:54.775 ","End":"03:00.920","Text":"B has no minimum because the infimum 0 does not belong to B."},{"Start":"03:01.380 ","End":"03:06.460","Text":"Now we saw that B had several upper bounds,"},{"Start":"03:06.460 ","End":"03:09.860","Text":"for example, 1, 4 and 40."},{"Start":"03:09.860 ","End":"03:12.645","Text":"But 1of them is special."},{"Start":"03:12.645 ","End":"03:14.910","Text":"That\u0027s the 1."},{"Start":"03:14.910 ","End":"03:20.100","Text":"That\u0027s because it\u0027s the smallest upper bound from all the upper bounds for B,"},{"Start":"03:20.100 ","End":"03:22.125","Text":"1 is the smallest."},{"Start":"03:22.125 ","End":"03:26.100","Text":"There\u0027s no upper bound of B less than 1."},{"Start":"03:26.100 ","End":"03:29.610","Text":"I mean, any upper bound has to at least be 1,"},{"Start":"03:29.610 ","End":"03:32.505","Text":"have to be bigger or equal to all of the elements."},{"Start":"03:32.505 ","End":"03:35.895","Text":"There\u0027s a name for a smallest upper bound."},{"Start":"03:35.895 ","End":"03:37.995","Text":"It\u0027s called a supremum."},{"Start":"03:37.995 ","End":"03:44.340","Text":"So 1 is the supremum of B and notation is sup, S-U-P."},{"Start":"03:44.340 ","End":"03:49.470","Text":"It\u0027s also called the least upper bound for obvious reasons."},{"Start":"03:49.470 ","End":"03:51.870","Text":"I will say in our case,"},{"Start":"03:51.870 ","End":"03:53.535","Text":"the supremum of B,"},{"Start":"03:53.535 ","End":"03:57.525","Text":"the least upper bound of B is 1 and it belongs to B."},{"Start":"03:57.525 ","End":"04:00.750","Text":"In general, if a supremum of the set belongs to the set,"},{"Start":"04:00.750 ","End":"04:03.165","Text":"it\u0027s also called the maximum of the set."},{"Start":"04:03.165 ","End":"04:05.310","Text":"This is what we have in our case."},{"Start":"04:05.310 ","End":"04:07.920","Text":"So 1, not only the supremum of B,"},{"Start":"04:07.920 ","End":"04:10.905","Text":"but it\u0027s the maximum of B because it belongs to B."},{"Start":"04:10.905 ","End":"04:13.755","Text":"You write max B equals 1."},{"Start":"04:13.755 ","End":"04:15.790","Text":"We defined 4 terms,"},{"Start":"04:15.790 ","End":"04:17.680","Text":"supremum maximum, infimum minimum,"},{"Start":"04:17.680 ","End":"04:18.970","Text":"but we did it informally."},{"Start":"04:18.970 ","End":"04:21.459","Text":"Let\u0027s make it a bit more formal."},{"Start":"04:21.459 ","End":"04:25.175","Text":"Given a set of real numbers, A,"},{"Start":"04:25.175 ","End":"04:30.905","Text":"if A has an upper bound which is smaller than any other upper bound of A,"},{"Start":"04:30.905 ","End":"04:33.635","Text":"it\u0027s called the supremum of A."},{"Start":"04:33.635 ","End":"04:35.510","Text":"If it happens to belong to A,"},{"Start":"04:35.510 ","End":"04:38.755","Text":"it\u0027s also called the maximum of A."},{"Start":"04:38.755 ","End":"04:42.920","Text":"Similarly, if A has a lower bound which is larger than any"},{"Start":"04:42.920 ","End":"04:47.150","Text":"other lower bound of A it\u0027s called the infimum of A."},{"Start":"04:47.150 ","End":"04:49.600","Text":"If it happens to belong to A,"},{"Start":"04:49.600 ","End":"04:52.210","Text":"it\u0027s called the minimum of A."},{"Start":"04:52.210 ","End":"04:53.855","Text":"That\u0027s basically it."},{"Start":"04:53.855 ","End":"04:59.960","Text":"But I\u0027d like to leave you with a couple of questions that we\u0027ll answer later."},{"Start":"04:59.960 ","End":"05:02.945","Text":"Does a set which is bounded from above,"},{"Start":"05:02.945 ","End":"05:06.140","Text":"necessarily have a least upper bound?"},{"Start":"05:06.140 ","End":"05:07.955","Text":"In other words, if it has an upper bound,"},{"Start":"05:07.955 ","End":"05:09.980","Text":"does it have at least 1?"},{"Start":"05:09.980 ","End":"05:14.460","Text":"Similarly, if a set is bounded from below, i.e."},{"Start":"05:14.460 ","End":"05:16.145","Text":"has a lower bound,"},{"Start":"05:16.145 ","End":"05:19.415","Text":"then does it have a greatest lower bound?"},{"Start":"05:19.415 ","End":"05:24.180","Text":"Like I said, we\u0027ll answer these later and we\u0027re done for now."}],"ID":26598},{"Watched":false,"Name":"Method for finding Supremum (L.U.B)","Duration":"8m 40s","ChapterTopicVideoID":25793,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.670","Text":"In this clip, I\u0027ll show you a method for finding the least upper bound of the set."},{"Start":"00:05.670 ","End":"00:07.920","Text":"It sometimes works, it\u0027s not always,"},{"Start":"00:07.920 ","End":"00:10.740","Text":"but it\u0027s a technique that could be useful."},{"Start":"00:10.740 ","End":"00:16.260","Text":"Least upper-bound is also known as supremum and abbreviated LUB."},{"Start":"00:16.260 ","End":"00:19.755","Text":"Suppose we are given a sequence a_n."},{"Start":"00:19.755 ","End":"00:22.290","Text":"Now remind you what a sequence is."},{"Start":"00:22.290 ","End":"00:25.334","Text":"Sequence is a function from the natural numbers,"},{"Start":"00:25.334 ","End":"00:26.970","Text":"in this case to the reals."},{"Start":"00:26.970 ","End":"00:30.090","Text":"Instead of writing a of n,"},{"Start":"00:30.090 ","End":"00:33.750","Text":"we write a sub n. Now,"},{"Start":"00:33.750 ","End":"00:36.900","Text":"sequence can give us a set."},{"Start":"00:36.900 ","End":"00:41.115","Text":"Just take all the a_n\u0027s and throw them into a set."},{"Start":"00:41.115 ","End":"00:44.140","Text":"This is a common way of getting a set from a sequence."},{"Start":"00:44.140 ","End":"00:47.315","Text":"Reason I grayed out the A naught is that"},{"Start":"00:47.315 ","End":"00:51.020","Text":"it depends whether the natural numbers include 0,"},{"Start":"00:51.020 ","End":"00:52.685","Text":"or doesn\u0027t include 0."},{"Start":"00:52.685 ","End":"00:54.680","Text":"This case, it doesn\u0027t matter."},{"Start":"00:54.680 ","End":"00:58.520","Text":"Now suppose we have a real number k. When can we"},{"Start":"00:58.520 ","End":"01:02.690","Text":"say that k is the least upper bound of the set a,"},{"Start":"01:02.690 ","End":"01:05.555","Text":"either that k is supremum of a."},{"Start":"01:05.555 ","End":"01:11.430","Text":"First thing to do is to show that k is an upper bound of a."},{"Start":"01:11.430 ","End":"01:14.290","Text":"We have to show that for all n in"},{"Start":"01:14.290 ","End":"01:19.730","Text":"n a_n is less than or equal to k. That makes it an upper bound,"},{"Start":"01:19.730 ","End":"01:21.710","Text":"but not necessarily the least."},{"Start":"01:21.710 ","End":"01:24.035","Text":"Now we subdivide into 2 cases."},{"Start":"01:24.035 ","End":"01:28.325","Text":"This number k could be an a and it could not be an a."},{"Start":"01:28.325 ","End":"01:30.935","Text":"If it is an a, that\u0027s the easy case."},{"Start":"01:30.935 ","End":"01:34.560","Text":"K is not only the supremum but the maximum."},{"Start":"01:34.560 ","End":"01:38.570","Text":"You can immediately say that k is the least upper bound of a."},{"Start":"01:38.570 ","End":"01:41.105","Text":"Let me explain why K is a least upper bound."},{"Start":"01:41.105 ","End":"01:42.470","Text":"Because k is in a,"},{"Start":"01:42.470 ","End":"01:45.120","Text":"it\u0027s equal to a_n for some n. Now,"},{"Start":"01:45.120 ","End":"01:49.785","Text":"suppose that k\u0027 is an upper bound of a,"},{"Start":"01:49.785 ","End":"01:53.335","Text":"then k\u0027 is bigger or equal to a_n."},{"Start":"01:53.335 ","End":"01:55.580","Text":"Because by definition of an upper bound,"},{"Start":"01:55.580 ","End":"01:58.115","Text":"it has to be bigger or equal to all the a_n."},{"Start":"01:58.115 ","End":"02:02.000","Text":"But a_n is equal to k. K\u0027 is bigger or"},{"Start":"02:02.000 ","End":"02:06.835","Text":"equal to k. Any upper bound is bigger or equal to k,"},{"Start":"02:06.835 ","End":"02:08.630","Text":"so k is the least."},{"Start":"02:08.630 ","End":"02:13.220","Text":"Now the next case is we\u0027re k does not belong to a."},{"Start":"02:13.220 ","End":"02:19.410","Text":"If we can show that for every S that\u0027s less than K,"},{"Start":"02:19.410 ","End":"02:22.975","Text":"there is some a_n which is bigger than s,"},{"Start":"02:22.975 ","End":"02:27.770","Text":"then we can conclude that k is the least upper bound of a."},{"Start":"02:27.770 ","End":"02:30.505","Text":"Now, to explain why this is so."},{"Start":"02:30.505 ","End":"02:32.685","Text":"Suppose that k\u0027,"},{"Start":"02:32.685 ","End":"02:35.100","Text":"which we\u0027ll also call s,"},{"Start":"02:35.100 ","End":"02:41.825","Text":"is less than k. Then a_n is bigger than k\u0027 for some n,"},{"Start":"02:41.825 ","End":"02:45.379","Text":"because we already agreed that we can show"},{"Start":"02:45.379 ","End":"02:51.060","Text":"that a_n is bigger than s for any s less than k. In particular,"},{"Start":"02:51.060 ","End":"02:53.110","Text":"a_n bigger than k\u0027."},{"Start":"02:53.720 ","End":"02:57.140","Text":"K\u0027 is not an upper bound."},{"Start":"02:57.140 ","End":"03:01.910","Text":"It\u0027s an upper bound, it has to be bigger or equal to every a_n."},{"Start":"03:01.910 ","End":"03:06.885","Text":"Any number less than k is not an upper bound."},{"Start":"03:06.885 ","End":"03:11.750","Text":"There\u0027s no upper bound less than k. K is the least upper bound."},{"Start":"03:11.750 ","End":"03:17.110","Text":"There\u0027s a diagram which you may or may not find of use. I won\u0027t go into it."},{"Start":"03:17.110 ","End":"03:20.880","Text":"Let\u0027s illustrate with 2 examples,"},{"Start":"03:20.880 ","End":"03:23.470","Text":"one for each case."},{"Start":"03:23.470 ","End":"03:30.930","Text":"Here\u0027s example 1, where a_n is n squared minus 8n plus 18 over n squared minus"},{"Start":"03:30.930 ","End":"03:38.435","Text":"8n plus 17 for all natural n. Let\u0027s say that the natural numbers don\u0027t include 0 here."},{"Start":"03:38.435 ","End":"03:41.480","Text":"We have to find the supremum of a."},{"Start":"03:41.480 ","End":"03:44.090","Text":"The first thing is to get"},{"Start":"03:44.090 ","End":"03:47.690","Text":"an impression of what\u0027s going on by writing out a few members of a."},{"Start":"03:47.690 ","End":"03:51.070","Text":"We\u0027ve written out A1 through A7,"},{"Start":"03:51.070 ","End":"03:54.635","Text":"11/10, 6/5,"},{"Start":"03:54.635 ","End":"03:57.045","Text":"3/2, 2/1. So on."},{"Start":"03:57.045 ","End":"03:59.730","Text":"In decimal, 1.1, 1.2,"},{"Start":"03:59.730 ","End":"04:03.435","Text":"1.5, 2 and 1.5, 1.2, 1.1."},{"Start":"04:03.435 ","End":"04:07.650","Text":"Looks like it increasing up to 2 and then going down again."},{"Start":"04:07.650 ","End":"04:10.640","Text":"We\u0027ll make an educated guess that the supremum of"},{"Start":"04:10.640 ","End":"04:14.900","Text":"a is 2 and that\u0027s the k that we\u0027re looking for."},{"Start":"04:14.900 ","End":"04:18.830","Text":"We are in case 1 because this k, which is 2,"},{"Start":"04:18.830 ","End":"04:22.060","Text":"is an element of the set a,"},{"Start":"04:22.060 ","End":"04:24.195","Text":"it\u0027s equal to A4."},{"Start":"04:24.195 ","End":"04:28.985","Text":"All we have to do is prove that 2 is an upper bound of a."},{"Start":"04:28.985 ","End":"04:33.410","Text":"In other words, that a_n is less than or equal to 2 for all n. We want to show that"},{"Start":"04:33.410 ","End":"04:39.515","Text":"this inequality holds for all n. That a_n is equal to this."},{"Start":"04:39.515 ","End":"04:46.100","Text":"Now, I claim that the denominator here is always positive."},{"Start":"04:46.100 ","End":"04:48.140","Text":"Several ways of showing this."},{"Start":"04:48.140 ","End":"04:51.860","Text":"One way would be to compute the discriminant of"},{"Start":"04:51.860 ","End":"04:57.050","Text":"this quadratic b squared minus 4ac turns out negative."},{"Start":"04:57.050 ","End":"04:58.940","Text":"It has no roots."},{"Start":"04:58.940 ","End":"05:02.240","Text":"It\u0027s never 0, so it\u0027s either always bigger than 0,"},{"Start":"05:02.240 ","End":"05:04.025","Text":"or always less than 0."},{"Start":"05:04.025 ","End":"05:07.775","Text":"Just plugging in n equals 0 and you see that it\u0027s positive."},{"Start":"05:07.775 ","End":"05:09.820","Text":"It\u0027s always positive."},{"Start":"05:09.820 ","End":"05:11.510","Text":"If it\u0027s always positive,"},{"Start":"05:11.510 ","End":"05:15.355","Text":"then we can multiply both sides of the inequality by it."},{"Start":"05:15.355 ","End":"05:20.360","Text":"This is what we get the numerator here and then this times 2 gives us this."},{"Start":"05:20.360 ","End":"05:25.640","Text":"Now we can throw everything to the right-hand side and just leave 0 on the left."},{"Start":"05:25.640 ","End":"05:27.455","Text":"This is what we get."},{"Start":"05:27.455 ","End":"05:30.080","Text":"Now the right-hand side is a perfect square."},{"Start":"05:30.080 ","End":"05:32.555","Text":"It\u0027s n minus 4 squared."},{"Start":"05:32.555 ","End":"05:35.125","Text":"Obviously, this will be true always."},{"Start":"05:35.125 ","End":"05:36.995","Text":"If this is always true,"},{"Start":"05:36.995 ","End":"05:38.960","Text":"then this is always true."},{"Start":"05:38.960 ","End":"05:41.540","Text":"That completes this example."},{"Start":"05:41.540 ","End":"05:43.760","Text":"Now I said do an example of each."},{"Start":"05:43.760 ","End":"05:47.750","Text":"Here\u0027s another example where k is not in the set a."},{"Start":"05:47.750 ","End":"05:52.335","Text":"We let a equal the set of n plus 2,"},{"Start":"05:52.335 ","End":"05:55.365","Text":"where n is a natural number."},{"Start":"05:55.365 ","End":"05:59.645","Text":"We have to find the supremum of the set a. Yeah,"},{"Start":"05:59.645 ","End":"06:02.520","Text":"this element is a_n."},{"Start":"06:02.560 ","End":"06:08.720","Text":"We write out a few of the members of a just to get a feeling 1/3,"},{"Start":"06:08.720 ","End":"06:11.390","Text":"2/4, 3/5 and so on."},{"Start":"06:11.390 ","End":"06:15.790","Text":"Denominator is 2 more than the numerator that\u0027s the pattern."},{"Start":"06:15.790 ","End":"06:20.570","Text":"Make an educated guess that the supremum is 1."},{"Start":"06:20.570 ","End":"06:22.820","Text":"These are all less than 1,"},{"Start":"06:22.820 ","End":"06:25.795","Text":"but very close to 1."},{"Start":"06:25.795 ","End":"06:28.200","Text":"1 is our k,"},{"Start":"06:28.200 ","End":"06:30.875","Text":"but 1 is not a member of the set a,"},{"Start":"06:30.875 ","End":"06:34.415","Text":"since n plus 2 will never equal 1."},{"Start":"06:34.415 ","End":"06:37.280","Text":"I mean, you can\u0027t have n equals n plus 2."},{"Start":"06:37.280 ","End":"06:38.555","Text":"That\u0027s impossible."},{"Start":"06:38.555 ","End":"06:42.865","Text":"In fact, n plus 2 is strictly less than 1,"},{"Start":"06:42.865 ","End":"06:47.910","Text":"which means that a_n are all less than k. This k,"},{"Start":"06:47.910 ","End":"06:50.110","Text":"which is 1, is an upper bound."},{"Start":"06:50.110 ","End":"06:53.620","Text":"We still have to show that it\u0027s a least upper bound."},{"Start":"06:53.620 ","End":"06:58.780","Text":"We do this by showing that if we take any s that\u0027s less than 1,"},{"Start":"06:58.780 ","End":"07:01.750","Text":"then for some n a_n is bigger than"},{"Start":"07:01.750 ","End":"07:06.970","Text":"s. A_n is this and we\u0027re looking for a solution to this inequality."},{"Start":"07:06.970 ","End":"07:09.655","Text":"We want to find n such that this is true."},{"Start":"07:09.655 ","End":"07:12.685","Text":"This is true if and only if,"},{"Start":"07:12.685 ","End":"07:15.955","Text":"I should say n plus 2 is positive."},{"Start":"07:15.955 ","End":"07:18.400","Text":"Of course, I mean n is a natural number,"},{"Start":"07:18.400 ","End":"07:20.695","Text":"so plus 2 is still positive."},{"Start":"07:20.695 ","End":"07:25.850","Text":"We can multiply both sides of the inequality and we get this."},{"Start":"07:25.850 ","End":"07:29.300","Text":"Then we can bring the ns over to the left,"},{"Start":"07:29.300 ","End":"07:32.345","Text":"take n out the brackets and we get this."},{"Start":"07:32.345 ","End":"07:41.160","Text":"Now 1 minus s is positive because s is less than 1. Yeah, here it is."},{"Start":"07:41.160 ","End":"07:49.900","Text":"We can divide both sides by 1 minus s and get n bigger than 2s/1 minus s. Now,"},{"Start":"07:49.900 ","End":"07:53.840","Text":"we haven\u0027t come to the Archimedean property yet."},{"Start":"07:53.840 ","End":"07:57.320","Text":"Just have to borrow that."},{"Start":"07:57.320 ","End":"08:03.065","Text":"What it says is that the set of natural numbers is unbounded."},{"Start":"08:03.065 ","End":"08:08.180","Text":"You can\u0027t have this number being bigger or equal to all natural numbers."},{"Start":"08:08.180 ","End":"08:12.950","Text":"There will be a natural number that\u0027s bigger than any positive real number."},{"Start":"08:12.950 ","End":"08:22.755","Text":"There is an n which is bigger than 2s/1 minus s. For this n a_n,"},{"Start":"08:22.755 ","End":"08:26.345","Text":"which is this is bigger than s as required."},{"Start":"08:26.345 ","End":"08:31.130","Text":"I wrote a note that what we\u0027re doing here is using the Archimedean property."},{"Start":"08:31.130 ","End":"08:33.365","Text":"We borrowed it from a future clip."},{"Start":"08:33.365 ","End":"08:36.770","Text":"Okay, that was all that we needed to show to complete"},{"Start":"08:36.770 ","End":"08:40.860","Text":"this example and that completes this clip."}],"ID":26597},{"Watched":false,"Name":"Example (Finding Sup\u0027)","Duration":"4m 1s","ChapterTopicVideoID":25796,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.435","Text":"This is a continuation really of the previous clip,"},{"Start":"00:04.435 ","End":"00:09.085","Text":"and we\u0027re just going to show another example of finding the least upper bound."},{"Start":"00:09.085 ","End":"00:14.230","Text":"This time, we\u0027ll let set AB set of all. Well, I won\u0027t read it out."},{"Start":"00:14.230 ","End":"00:17.965","Text":"This expression for N belongs to N. Yeah,"},{"Start":"00:17.965 ","End":"00:20.630","Text":"I meant to label this as AN."},{"Start":"00:20.630 ","End":"00:24.530","Text":"We have to find the supremum of the set A."},{"Start":"00:24.530 ","End":"00:30.325","Text":"Start out by writing a few members of the set A just to get an idea."},{"Start":"00:30.325 ","End":"00:32.830","Text":"Here it is, and in decimal."},{"Start":"00:32.830 ","End":"00:37.600","Text":"It looks like the least upper bound is a halt."},{"Start":"00:37.600 ","End":"00:39.470","Text":"This, That\u0027s an educated guess."},{"Start":"00:39.470 ","End":"00:42.735","Text":"Let\u0027s go with that and see if we can prove it."},{"Start":"00:42.735 ","End":"00:46.010","Text":"First thing to do is to show that it\u0027s an upper bound."},{"Start":"00:46.010 ","End":"00:48.545","Text":"Afterwards, we\u0027ll show that it\u0027s the least."},{"Start":"00:48.545 ","End":"00:51.890","Text":"We need to show that for each element in A, in other words,"},{"Start":"00:51.890 ","End":"00:53.135","Text":"for all AN,"},{"Start":"00:53.135 ","End":"00:57.035","Text":"AN is less than or equal to 1 half."},{"Start":"00:57.035 ","End":"01:00.605","Text":"In fact, I can even show that it\u0027s strictly less than a half."},{"Start":"01:00.605 ","End":"01:02.060","Text":"And I was proving this,"},{"Start":"01:02.060 ","End":"01:03.784","Text":"I noticed that we could go further."},{"Start":"01:03.784 ","End":"01:06.290","Text":"This is useful. So let\u0027s see."},{"Start":"01:06.290 ","End":"01:13.790","Text":"This is our typical AN and this will be less than a half if and only if,"},{"Start":"01:13.790 ","End":"01:16.965","Text":"the denominator is bigger than 0."},{"Start":"01:16.965 ","End":"01:20.330","Text":"We can multiply out by the denominator and by 2,"},{"Start":"01:20.330 ","End":"01:22.265","Text":"in other words, cross-multiply."},{"Start":"01:22.265 ","End":"01:23.690","Text":"And we get 2 times,"},{"Start":"01:23.690 ","End":"01:26.255","Text":"this is less than this times this."},{"Start":"01:26.255 ","End":"01:29.375","Text":"This is clearly positive because N is positive."},{"Start":"01:29.375 ","End":"01:31.880","Text":"Now here and here we have common part."},{"Start":"01:31.880 ","End":"01:34.745","Text":"This is the same here and here."},{"Start":"01:34.745 ","End":"01:40.955","Text":"We can subtract it from both sides and end up with 2 is less than 4."},{"Start":"01:40.955 ","End":"01:42.530","Text":"Since this is true,"},{"Start":"01:42.530 ","End":"01:45.185","Text":"this is true and therefore this is true."},{"Start":"01:45.185 ","End":"01:49.220","Text":"We\u0027ve shown that AN is less than a half."},{"Start":"01:49.220 ","End":"01:51.740","Text":"Remember in the previous clip we had 2 cases."},{"Start":"01:51.740 ","End":"01:54.740","Text":"1 where this, what we called K is in"},{"Start":"01:54.740 ","End":"01:59.405","Text":"the set A and 1 where it\u0027s not in the set A and 1 was easy when it\u0027s in the set A."},{"Start":"01:59.405 ","End":"02:02.060","Text":"Unfortunately, we\u0027re in the difficult case."},{"Start":"02:02.060 ","End":"02:04.910","Text":"A half does not belong to A."},{"Start":"02:04.910 ","End":"02:07.520","Text":"Show this, well, it turns out to be easy"},{"Start":"02:07.520 ","End":"02:10.610","Text":"because we showed that AN is strictly less than a half,"},{"Start":"02:10.610 ","End":"02:12.695","Text":"which means that it\u0027s not equal to a half."},{"Start":"02:12.695 ","End":"02:15.170","Text":"So none of the AN is equal to a half."},{"Start":"02:15.170 ","End":"02:17.450","Text":"If I hadn\u0027t used the shortcut,"},{"Start":"02:17.450 ","End":"02:18.920","Text":"we could have done it the long way,"},{"Start":"02:18.920 ","End":"02:20.795","Text":"which is essentially a repetition of this."},{"Start":"02:20.795 ","End":"02:24.320","Text":"Say this equals a half if and only if this equals this,"},{"Start":"02:24.320 ","End":"02:26.570","Text":"and if and only if 2 equals 4."},{"Start":"02:26.570 ","End":"02:29.095","Text":"And this certainly is not true."},{"Start":"02:29.095 ","End":"02:33.860","Text":"So, this is also not true for any N. Either"},{"Start":"02:33.860 ","End":"02:38.870","Text":"way we\u0027ve shown that half is an upper bound and it\u0027s not in the set."},{"Start":"02:38.870 ","End":"02:43.175","Text":"We have to do some work to show that it\u0027s the least upper bound."},{"Start":"02:43.175 ","End":"02:46.400","Text":"Basically, what we have to do is to show that"},{"Start":"02:46.400 ","End":"02:49.565","Text":"if we have some S which is less than a half,"},{"Start":"02:49.565 ","End":"02:52.460","Text":"then at least 1 of the AN will be bigger than"},{"Start":"02:52.460 ","End":"02:55.460","Text":"this S. This is what we discussed in the previous clip."},{"Start":"02:55.460 ","End":"02:59.220","Text":"I\u0027m going to let Epsilon equal a half minus S,"},{"Start":"02:59.220 ","End":"03:01.685","Text":"this just makes work easier."},{"Start":"03:01.685 ","End":"03:02.990","Text":"Instead of working with AN,"},{"Start":"03:02.990 ","End":"03:05.300","Text":"I\u0027ll work with a half minus AN."},{"Start":"03:05.300 ","End":"03:12.380","Text":"AN is bigger than S if and only if a half minus AN is less than a half minus S,"},{"Start":"03:12.380 ","End":"03:15.790","Text":"and that is equal to Epsilon."},{"Start":"03:15.790 ","End":"03:20.300","Text":"So yeah, I teach you how to work with a half minus N you\u0027ll see the expression comes out"},{"Start":"03:20.300 ","End":"03:25.010","Text":"simpler because a half minus AN comes out to be this."},{"Start":"03:25.010 ","End":"03:28.130","Text":"We want this to be less than Epsilon to show that at least"},{"Start":"03:28.130 ","End":"03:31.165","Text":"1N makes this less than Epsilon."},{"Start":"03:31.165 ","End":"03:34.070","Text":"Now this is less than Epsilon if and only if the"},{"Start":"03:34.070 ","End":"03:38.125","Text":"reciprocal is bigger than 1 over Epsilon."},{"Start":"03:38.125 ","End":"03:44.240","Text":"Now we can choose N to be bigger than 1 over Epsilon by the Archimedean property."},{"Start":"03:44.240 ","End":"03:47.660","Text":"And if N is bigger than 1 over Epsilon,"},{"Start":"03:47.660 ","End":"03:50.120","Text":"and certainly this thing,"},{"Start":"03:50.120 ","End":"03:55.235","Text":"2M squared plus 2M plus 4 is bigger than 1 over Epsilon because this is bigger than N,"},{"Start":"03:55.235 ","End":"03:57.290","Text":"and this is as required."},{"Start":"03:57.290 ","End":"03:59.929","Text":"And that\u0027s the last step we needed."},{"Start":"03:59.929 ","End":"04:02.940","Text":"And so we are done."}],"ID":26600},{"Watched":false,"Name":"Method for finding Infimum (G.L.B)","Duration":"3m 59s","ChapterTopicVideoID":25792,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.090","Text":"We\u0027re continuing the previous couple of clips"},{"Start":"00:03.090 ","End":"00:05.880","Text":"where we learned to find the least upper bound,"},{"Start":"00:05.880 ","End":"00:09.300","Text":"and this time we\u0027ll discuss the greatest lower bound."},{"Start":"00:09.300 ","End":"00:11.820","Text":"It\u0027s very similar, just opposite,"},{"Start":"00:11.820 ","End":"00:15.525","Text":"which is lower bound which is also the infimum."},{"Start":"00:15.525 ","End":"00:21.300","Text":"Like before, we\u0027ll be working with a set that comes from a sequence a_n,"},{"Start":"00:21.300 ","End":"00:23.265","Text":"where n is a natural number;"},{"Start":"00:23.265 ","End":"00:24.840","Text":"with or without the a_0,"},{"Start":"00:24.840 ","End":"00:26.985","Text":"it doesn\u0027t really matter."},{"Start":"00:26.985 ","End":"00:29.325","Text":"Suppose we have a lower bound,"},{"Start":"00:29.325 ","End":"00:31.305","Text":"k for A,"},{"Start":"00:31.305 ","End":"00:37.310","Text":"when can we say that k is the greatest lower bound or infimum of a?"},{"Start":"00:37.310 ","End":"00:40.160","Text":"As before, we\u0027ll have 2 cases where k is in"},{"Start":"00:40.160 ","End":"00:44.595","Text":"the set A and then when k is not in the set A,"},{"Start":"00:44.595 ","End":"00:46.200","Text":"and this will be the easy k,"},{"Start":"00:46.200 ","End":"00:50.360","Text":"it\u0027s like before, because this means that k is the minimum."},{"Start":"00:50.360 ","End":"00:54.095","Text":"Whenever you have a lower bound that belongs to the set,"},{"Start":"00:54.095 ","End":"00:55.475","Text":"then it\u0027s the minimum,"},{"Start":"00:55.475 ","End":"01:00.490","Text":"and then you can immediately say that k is the least upper bound."},{"Start":"01:00.490 ","End":"01:02.505","Text":"The reason is,"},{"Start":"01:02.505 ","End":"01:04.290","Text":"that k belongs to A,"},{"Start":"01:04.290 ","End":"01:06.175","Text":"so k is 1 of the a_n."},{"Start":"01:06.175 ","End":"01:10.415","Text":"If k prime is the lower bound of A also,"},{"Start":"01:10.415 ","End":"01:14.525","Text":"then k prime has to be less than or equal to a_n because it\u0027s"},{"Start":"01:14.525 ","End":"01:19.505","Text":"a lower bound and this is equal to k. So k prime is less than or equal to k,"},{"Start":"01:19.505 ","End":"01:22.070","Text":"so all the lower bounds are less than or equal to k,"},{"Start":"01:22.070 ","End":"01:24.865","Text":"so k is the greatest lower bound."},{"Start":"01:24.865 ","End":"01:28.910","Text":"Now, case 2, where k does not belong to the set A,"},{"Start":"01:28.910 ","End":"01:32.540","Text":"and then it becomes the greatest lower bound if we can"},{"Start":"01:32.540 ","End":"01:36.395","Text":"show the following: that for all s that\u0027s bigger than k,"},{"Start":"01:36.395 ","End":"01:38.890","Text":"there is 1 of the a_n at least,"},{"Start":"01:38.890 ","End":"01:43.790","Text":"such that a_n is less than s. If we can show this,"},{"Start":"01:43.790 ","End":"01:46.480","Text":"then k is the greatest lower bound of A."},{"Start":"01:46.480 ","End":"01:55.385","Text":"The reason is that suppose you have k prime which is a greater lower bound than k,"},{"Start":"01:55.385 ","End":"02:01.755","Text":"then a_n is less than k prime because k prime is like our s for some n,"},{"Start":"02:01.755 ","End":"02:08.150","Text":"and so k prime is not a lower bound because 1 of the members of the set is less than it."},{"Start":"02:08.150 ","End":"02:12.065","Text":"There\u0027s no lower bound greater than k. Now,"},{"Start":"02:12.065 ","End":"02:14.120","Text":"let\u0027s go and do an example."},{"Start":"02:14.120 ","End":"02:17.645","Text":"This is a picture that may or may not help to visualize."},{"Start":"02:17.645 ","End":"02:20.405","Text":"This is Example 4 because we\u0027re continuing."},{"Start":"02:20.405 ","End":"02:24.740","Text":"We let a equal 1 over n^4 plus n squared plus 4,"},{"Start":"02:24.740 ","End":"02:27.275","Text":"and we have to find the supremum, i.e."},{"Start":"02:27.275 ","End":"02:29.799","Text":"the greatest lower bound."},{"Start":"02:29.799 ","End":"02:33.440","Text":"Start off as usual by writing a few members of A."},{"Start":"02:33.440 ","End":"02:36.500","Text":"We\u0027ll assume that natural number start from 1,"},{"Start":"02:36.500 ","End":"02:39.205","Text":"so 1 over 1 plus 1 plus 4 is 6."},{"Start":"02:39.205 ","End":"02:42.165","Text":"Let me get 1 over 24, 1 over 94."},{"Start":"02:42.165 ","End":"02:48.825","Text":"What it looks like is that this is heading towards 0 but staying positive."},{"Start":"02:48.825 ","End":"02:52.910","Text":"Make an educated guess that the infimum is 0,"},{"Start":"02:52.910 ","End":"03:02.985","Text":"that\u0027s our k. Note that 0 is not in the set because this is 1 over a positive number,"},{"Start":"03:02.985 ","End":"03:05.160","Text":"and that can\u0027t be equal to 0,"},{"Start":"03:05.160 ","End":"03:07.215","Text":"1 over anything can\u0027t be 0."},{"Start":"03:07.215 ","End":"03:08.630","Text":"What it remains to show is,"},{"Start":"03:08.630 ","End":"03:10.790","Text":"like we said moment earlier,"},{"Start":"03:10.790 ","End":"03:14.070","Text":"we have to show that if s is bigger than 0,"},{"Start":"03:14.070 ","End":"03:16.580","Text":"bigger than k which is 0 in this case,"},{"Start":"03:16.580 ","End":"03:18.590","Text":"then for some n,"},{"Start":"03:18.590 ","End":"03:21.175","Text":"a_n is less than s,"},{"Start":"03:21.175 ","End":"03:24.210","Text":"and a_n is this."},{"Start":"03:24.210 ","End":"03:29.690","Text":"We want to find at least 1 n for which this is less than s. Now it\u0027s sufficient to"},{"Start":"03:29.690 ","End":"03:36.380","Text":"find n such that this is bigger than 1 over s. If you take the reciprocals,"},{"Start":"03:36.380 ","End":"03:39.650","Text":"then you reverse the inequality."},{"Start":"03:39.650 ","End":"03:43.385","Text":"But if we choose n bigger than 1 over s,"},{"Start":"03:43.385 ","End":"03:46.670","Text":"and we can by the Archimedean property,"},{"Start":"03:46.670 ","End":"03:50.855","Text":"then this, which is bigger than n,"},{"Start":"03:50.855 ","End":"03:55.520","Text":"will also be bigger than 1 over s as required."},{"Start":"03:55.520 ","End":"04:00.090","Text":"This completes the exercise and we\u0027re done."}],"ID":26596},{"Watched":false,"Name":"Exercise 1","Duration":"1m 55s","ChapterTopicVideoID":25797,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.555","Text":"In this exercise, we define the set C to be the set of all real numbers x,"},{"Start":"00:06.555 ","End":"00:11.700","Text":"such that x squared minus 4 over x minus 2 squared is less than or equal to 0."},{"Start":"00:11.700 ","End":"00:18.780","Text":"Now our task is to show that the set C is bounded and to find its infimum and supremum."},{"Start":"00:18.780 ","End":"00:24.015","Text":"The domain of this function is everything except x equals 2,"},{"Start":"00:24.015 ","End":"00:27.360","Text":"because that will cause a problem of dividing by 0,"},{"Start":"00:27.360 ","End":"00:31.095","Text":"so x not equals 2 is the domain."},{"Start":"00:31.095 ","End":"00:35.235","Text":"Note that within a domain that denominator is positive."},{"Start":"00:35.235 ","End":"00:39.005","Text":"We can simplify what C is and say that"},{"Start":"00:39.005 ","End":"00:44.060","Text":"the numerator has to be non-negative because we\u0027re dividing by a positive thing."},{"Start":"00:44.060 ","End":"00:50.195","Text":"But also the x has to be within the domain x not equal to 2. We get this."},{"Start":"00:50.195 ","End":"00:52.940","Text":"Now, we can solve this inequality to"},{"Start":"00:52.940 ","End":"00:57.605","Text":"quadratic inequality and here\u0027s a picture of the parabola,"},{"Start":"00:57.605 ","End":"00:59.825","Text":"y equals x squared minus 4,"},{"Start":"00:59.825 ","End":"01:04.495","Text":"which is less than or equal to 0 is this part here."},{"Start":"01:04.495 ","End":"01:10.390","Text":"Now it includes the minus 2 and it would include the 2 except"},{"Start":"01:10.390 ","End":"01:17.375","Text":"we\u0027ve explicitly forbidden x equals 2 because the denominator here would be 0."},{"Start":"01:17.375 ","End":"01:21.230","Text":"Just write this as x between minus 2 and 2 inclusive,"},{"Start":"01:21.230 ","End":"01:23.635","Text":"but x is not equal to 2."},{"Start":"01:23.635 ","End":"01:29.690","Text":"That means that we just have to change this less than or equal to 2 to a less than here."},{"Start":"01:29.690 ","End":"01:33.290","Text":"You could write this in interval notation as the interval from"},{"Start":"01:33.290 ","End":"01:37.130","Text":"minus 2 to 2 half-closed on the left side."},{"Start":"01:37.130 ","End":"01:39.005","Text":"Now for this interval,"},{"Start":"01:39.005 ","End":"01:42.095","Text":"the infimum and the minimum is minus 2."},{"Start":"01:42.095 ","End":"01:44.929","Text":"Whenever you have a minimum in the set,"},{"Start":"01:44.929 ","End":"01:46.175","Text":"it\u0027s also an infimum,"},{"Start":"01:46.175 ","End":"01:49.895","Text":"but there is no maximum because it\u0027s open on the right,"},{"Start":"01:49.895 ","End":"01:51.200","Text":"but there is a supremum."},{"Start":"01:51.200 ","End":"01:56.880","Text":"The supremum is 2 and that\u0027s the conclusion of this exercise."}],"ID":26601},{"Watched":false,"Name":"Exercise 2","Duration":"2m 32s","ChapterTopicVideoID":25798,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.265","Text":"This exercise has 2 parts,"},{"Start":"00:02.265 ","End":"00:04.470","Text":"each concerning the supremum."},{"Start":"00:04.470 ","End":"00:11.745","Text":"In the first part, we have a set of numbers s. We have to prove that if s has a supremum,"},{"Start":"00:11.745 ","End":"00:13.860","Text":"also known as the least upper bound,"},{"Start":"00:13.860 ","End":"00:15.585","Text":"then this is unique."},{"Start":"00:15.585 ","End":"00:20.535","Text":"In part b, we have to prove that the empty set doesn\u0027t have a supremum."},{"Start":"00:20.535 ","End":"00:26.235","Text":"Now in part a, it\u0027s almost trivial because they can only be 1 least,"},{"Start":"00:26.235 ","End":"00:27.990","Text":"but let\u0027s write it formally."},{"Start":"00:27.990 ","End":"00:30.780","Text":"Let\u0027s say that m1 and m2 are"},{"Start":"00:30.780 ","End":"00:35.514","Text":"2 least upper bounds for s. We have to show that they\u0027re actually the same."},{"Start":"00:35.514 ","End":"00:40.250","Text":"Now, a least upper bound is in particular an upper bound,"},{"Start":"00:40.250 ","End":"00:42.875","Text":"m1 is a least upper bound,"},{"Start":"00:42.875 ","End":"00:48.265","Text":"and m2 is an upper bound because it\u0027s the least upper bound and particular upper bound."},{"Start":"00:48.265 ","End":"00:51.080","Text":"The least is always smaller than any other,"},{"Start":"00:51.080 ","End":"00:55.490","Text":"m1 has to be less than or equal to m2 and if you look at it the other way"},{"Start":"00:55.490 ","End":"01:00.140","Text":"around m2 is a least upper bound and m1 is an upper bound,"},{"Start":"01:00.140 ","End":"01:02.875","Text":"m2 is less than or equal to m1."},{"Start":"01:02.875 ","End":"01:07.459","Text":"From these 2, we get that m1 equals m2."},{"Start":"01:07.459 ","End":"01:10.070","Text":"Now the empty set is a bit tricky because"},{"Start":"01:10.070 ","End":"01:12.710","Text":"it uses some property of logic. Well, you\u0027ll see."},{"Start":"01:12.710 ","End":"01:17.860","Text":"The claim is that any number is an upper bound for the empty set,"},{"Start":"01:17.860 ","End":"01:20.875","Text":"this is what we call vacuously true."},{"Start":"01:20.875 ","End":"01:27.830","Text":"Any x in the empty set is less than or equal to m. Might say,"},{"Start":"01:27.830 ","End":"01:29.855","Text":"Yeah, but there is no x in the empty set."},{"Start":"01:29.855 ","End":"01:32.495","Text":"Well, that\u0027s exactly what makes it vacuously true."},{"Start":"01:32.495 ","End":"01:37.460","Text":"See in logic, a implies b is true if a is false,"},{"Start":"01:37.460 ","End":"01:39.665","Text":"and this is always false,"},{"Start":"01:39.665 ","End":"01:43.860","Text":"this is true of every m and r is an upper bound."},{"Start":"01:43.860 ","End":"01:46.610","Text":"The set of upper bounds of the empty set is all of"},{"Start":"01:46.610 ","End":"01:50.765","Text":"the reals and there\u0027s no least real number."},{"Start":"01:50.765 ","End":"01:56.470","Text":"That proves that the empty set has no least upper bound and we\u0027re done,"},{"Start":"01:56.470 ","End":"01:57.920","Text":"but don\u0027t go just yet."},{"Start":"01:57.920 ","End":"01:59.330","Text":"I want to make a remark."},{"Start":"01:59.330 ","End":"02:04.010","Text":"Notice that this exercise had 2 parts each involving the supremum."},{"Start":"02:04.010 ","End":"02:06.895","Text":"Supremum is unique and empty set doesn\u0027t have a supremum."},{"Start":"02:06.895 ","End":"02:10.610","Text":"We could change the word supremum for infimum at"},{"Start":"02:10.610 ","End":"02:14.405","Text":"least upper bound for greatest lower bound and also have a claim."},{"Start":"02:14.405 ","End":"02:18.245","Text":"What I\u0027m saying is, given a set of numbers,"},{"Start":"02:18.245 ","End":"02:20.255","Text":"if it has an infimum,"},{"Start":"02:20.255 ","End":"02:25.575","Text":"then it\u0027s unique and the empty set doesn\u0027t have an infimum."},{"Start":"02:25.575 ","End":"02:33.030","Text":"1 proof is it\u0027s entirely similar to what we just proved for supremum We\u0027re done."}],"ID":26602},{"Watched":false,"Name":"Exercise 3","Duration":"2m 24s","ChapterTopicVideoID":25776,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"This exercise is in 2 parts,"},{"Start":"00:02.490 ","End":"00:03.840","Text":"1 involving the supremum,"},{"Start":"00:03.840 ","End":"00:05.400","Text":"1 involving the infimum."},{"Start":"00:05.400 ","End":"00:11.355","Text":"Given a set A with supremum Alpha and given a positive Epsilon,"},{"Start":"00:11.355 ","End":"00:14.790","Text":"we have to show that there is some x in A,"},{"Start":"00:14.790 ","End":"00:19.020","Text":"which satisfies this inequality that x is less than or equal to Alpha,"},{"Start":"00:19.020 ","End":"00:22.260","Text":"but bigger than Alpha minus Epsilon."},{"Start":"00:22.260 ","End":"00:25.725","Text":"Part b is the analogous statement with infimum."},{"Start":"00:25.725 ","End":"00:30.360","Text":"Given a set A with infimum Beta and given a positive Epsilon,"},{"Start":"00:30.360 ","End":"00:32.810","Text":"we have to show that there is some x in A,"},{"Start":"00:32.810 ","End":"00:35.240","Text":"such that this inequality holds,"},{"Start":"00:35.240 ","End":"00:37.445","Text":"x is bigger or equal to Beta,"},{"Start":"00:37.445 ","End":"00:40.415","Text":"but strictly less than Beta plus Epsilon."},{"Start":"00:40.415 ","End":"00:42.515","Text":"Now I\u0027m going to simplify the problem."},{"Start":"00:42.515 ","End":"00:44.375","Text":"We only have to really prove half of it."},{"Start":"00:44.375 ","End":"00:46.225","Text":"If x is in A,"},{"Start":"00:46.225 ","End":"00:51.560","Text":"then any element of A is less than or equal to the supremum of the set, which is Alpha."},{"Start":"00:51.560 ","End":"00:53.855","Text":"This is already guaranteed."},{"Start":"00:53.855 ","End":"01:00.290","Text":"We just have to find an x that satisfies this part because this part\u0027s guaranteed."},{"Start":"01:00.290 ","End":"01:03.350","Text":"We\u0027ll do a proof by contradiction by supposing that"},{"Start":"01:03.350 ","End":"01:07.295","Text":"there is no such x which satisfies this."},{"Start":"01:07.295 ","End":"01:09.065","Text":"If it doesn\u0027t satisfy this,"},{"Start":"01:09.065 ","End":"01:15.000","Text":"then it\u0027s less than or equal to Alpha minus Epsilon for all x in A."},{"Start":"01:15.000 ","End":"01:19.490","Text":"That means that Alpha minus Epsilon is an upper bound."},{"Start":"01:19.490 ","End":"01:22.250","Text":"Because if every x is less than or equal to this,"},{"Start":"01:22.250 ","End":"01:28.340","Text":"then this is an upper bound of A that\u0027s smaller than the least upper bound Alpha."},{"Start":"01:28.340 ","End":"01:30.680","Text":"Well, it can\u0027t be smaller than the least,"},{"Start":"01:30.680 ","End":"01:34.205","Text":"so that contradiction proves our claim."},{"Start":"01:34.205 ","End":"01:37.625","Text":"The other way with the infimum is entirely similar."},{"Start":"01:37.625 ","End":"01:40.700","Text":"Like I said, if x is in A,"},{"Start":"01:40.700 ","End":"01:45.710","Text":"then automatically it\u0027s bigger or equal to the infimum."},{"Start":"01:45.710 ","End":"01:50.735","Text":"We just have to find x that satisfies the other half of the inequality,"},{"Start":"01:50.735 ","End":"01:54.485","Text":"that x is less than Beta plus Epsilon."},{"Start":"01:54.485 ","End":"01:57.415","Text":"As before, we\u0027ll do it by contradiction."},{"Start":"01:57.415 ","End":"01:59.675","Text":"If there\u0027s no x less than,"},{"Start":"01:59.675 ","End":"02:05.215","Text":"for every x it\u0027s bigger or equal to Beta plus Epsilon for every x in A."},{"Start":"02:05.215 ","End":"02:10.855","Text":"That means that Beta plus Epsilon is a lower bound of A,"},{"Start":"02:10.855 ","End":"02:12.620","Text":"but it\u0027s bigger than Beta,"},{"Start":"02:12.620 ","End":"02:14.450","Text":"which is the greatest lower bound."},{"Start":"02:14.450 ","End":"02:18.145","Text":"A lower bound can\u0027t be greater than the greatest,"},{"Start":"02:18.145 ","End":"02:22.430","Text":"so that\u0027s a contradiction and that proves Part b,"},{"Start":"02:22.430 ","End":"02:24.540","Text":"and so we\u0027re done."}],"ID":26580},{"Watched":false,"Name":"Exercise 4 - Density of Reals","Duration":"6m 28s","ChapterTopicVideoID":25777,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"This exercise has several parts."},{"Start":"00:03.075 ","End":"00:07.769","Text":"In part a, we have to show that between any 2 real numbers,"},{"Start":"00:07.769 ","End":"00:09.645","Text":"there is a 3rd real number."},{"Start":"00:09.645 ","End":"00:13.935","Text":"This property is called density of the real numbers."},{"Start":"00:13.935 ","End":"00:18.345","Text":"B through E are all related to maximum and supremum."},{"Start":"00:18.345 ","End":"00:21.495","Text":"For non-empty set S has a maximum,"},{"Start":"00:21.495 ","End":"00:23.040","Text":"then it has a supremum,"},{"Start":"00:23.040 ","End":"00:27.090","Text":"and the 2 are equal intervals of the following form,"},{"Start":"00:27.090 ","End":"00:29.355","Text":"they have no maximum."},{"Start":"00:29.355 ","End":"00:31.245","Text":"One of these forms,"},{"Start":"00:31.245 ","End":"00:32.790","Text":"the same as here,"},{"Start":"00:32.790 ","End":"00:34.555","Text":"they do have a supremum,"},{"Start":"00:34.555 ","End":"00:36.400","Text":"and B is the supremum."},{"Start":"00:36.400 ","End":"00:40.170","Text":"Intervals of this form have no supremum."},{"Start":"00:40.170 ","End":"00:42.315","Text":"We\u0027ll start with part a."},{"Start":"00:42.315 ","End":"00:44.985","Text":"Now, a and b are distinct."},{"Start":"00:44.985 ","End":"00:49.670","Text":"We can assume without loss of generality that a is less than b."},{"Start":"00:49.670 ","End":"00:52.660","Text":"If it\u0027s the other way around, then just relabel them."},{"Start":"00:52.660 ","End":"00:56.210","Text":"Now, let c be the average of a and b."},{"Start":"00:56.210 ","End":"01:00.710","Text":"The claim is that c is between a and b,"},{"Start":"01:00.710 ","End":"01:03.755","Text":"bigger than a and less than b. Here\u0027s the proof."},{"Start":"01:03.755 ","End":"01:05.420","Text":"A is less than b,"},{"Start":"01:05.420 ","End":"01:10.400","Text":"so a plus a is less than a plus b because I\u0027ve swapped an a for a b."},{"Start":"01:10.400 ","End":"01:12.530","Text":"Now I swap the other a for a b."},{"Start":"01:12.530 ","End":"01:15.050","Text":"These 3 quantities are in this order."},{"Start":"01:15.050 ","End":"01:16.715","Text":"What this means is 2a,"},{"Start":"01:16.715 ","End":"01:19.970","Text":"is less than a plus b is less than 2b, now divide by 2,"},{"Start":"01:19.970 ","End":"01:23.510","Text":"and we get that a is less than a plus b over 2,"},{"Start":"01:23.510 ","End":"01:25.570","Text":"which is less than b."},{"Start":"01:25.570 ","End":"01:30.255","Text":"This is what we call c. So a is less than c is less than b."},{"Start":"01:30.255 ","End":"01:33.260","Text":"That proves the claim that between a and b,"},{"Start":"01:33.260 ","End":"01:35.330","Text":"there is a 3rd real number,"},{"Start":"01:35.330 ","End":"01:38.680","Text":"and that\u0027s our c. Onto part b,"},{"Start":"01:38.680 ","End":"01:41.360","Text":"I\u0027ll paraphrase the question, because it\u0027s disappeared."},{"Start":"01:41.360 ","End":"01:43.490","Text":"Basically, it says if you have a maximum,"},{"Start":"01:43.490 ","End":"01:44.935","Text":"it\u0027s also a supremum."},{"Start":"01:44.935 ","End":"01:50.225","Text":"Let m be the maximum of the set s. Now by the definition of maximum,"},{"Start":"01:50.225 ","End":"01:54.590","Text":"it belongs to the set and it\u0027s an upper bound for the set."},{"Start":"01:54.590 ","End":"01:58.610","Text":"Suppose that m is not the least upper bound,"},{"Start":"01:58.610 ","End":"02:01.040","Text":"not a supremum is not the least upper bound,"},{"Start":"02:01.040 ","End":"02:04.820","Text":"so the supremum of s is less than m,"},{"Start":"02:04.820 ","End":"02:08.775","Text":"we\u0027ll call it s. Since m belongs to the set s,"},{"Start":"02:08.775 ","End":"02:10.880","Text":"it\u0027s less than or equal to the supremum,"},{"Start":"02:10.880 ","End":"02:12.905","Text":"less than or equal to the least upper bound,"},{"Start":"02:12.905 ","End":"02:15.260","Text":"which is s. I claim we have a contradiction."},{"Start":"02:15.260 ","End":"02:19.355","Text":"Look, s is less than m. On the other hand,"},{"Start":"02:19.355 ","End":"02:25.055","Text":"m is less than or equal to s. We have a contradiction,"},{"Start":"02:25.055 ","End":"02:27.890","Text":"and the contradiction came from assuming that"},{"Start":"02:27.890 ","End":"02:31.250","Text":"m is not the least upper bound, not the supremum."},{"Start":"02:31.250 ","End":"02:32.629","Text":"It is the supremum."},{"Start":"02:32.629 ","End":"02:38.210","Text":"Now on to part c. I is 1 of these types of intervals."},{"Start":"02:38.210 ","End":"02:41.585","Text":"We have to show that I doesn\u0027t have a maximum."},{"Start":"02:41.585 ","End":"02:43.595","Text":"We\u0027ll do that by contradiction."},{"Start":"02:43.595 ","End":"02:46.130","Text":"Suppose that the interval I does have a maximum,"},{"Start":"02:46.130 ","End":"02:49.835","Text":"call it m. By definition of a maximum,"},{"Start":"02:49.835 ","End":"02:53.275","Text":"this m is in the set, in the interval."},{"Start":"02:53.275 ","End":"02:55.290","Text":"If it\u0027s in the interval,"},{"Start":"02:55.290 ","End":"02:56.820","Text":"in either of these 3 cases,"},{"Start":"02:56.820 ","End":"02:58.515","Text":"it\u0027s got to be less than b."},{"Start":"02:58.515 ","End":"03:01.775","Text":"By the density of r, in other words, part a,"},{"Start":"03:01.775 ","End":"03:07.085","Text":"we can find a real number c that\u0027s between m and b,"},{"Start":"03:07.085 ","End":"03:08.990","Text":"m less than c, less than b."},{"Start":"03:08.990 ","End":"03:11.555","Text":"Now, c is in the interval,"},{"Start":"03:11.555 ","End":"03:13.760","Text":"because m is in the interval."},{"Start":"03:13.760 ","End":"03:15.545","Text":"If I take it further right,"},{"Start":"03:15.545 ","End":"03:19.540","Text":"but still less than b and it\u0027s still in the interval."},{"Start":"03:19.540 ","End":"03:22.030","Text":"It\u0027s in the interval I,"},{"Start":"03:22.030 ","End":"03:23.710","Text":"but bigger than the maximum,"},{"Start":"03:23.710 ","End":"03:26.095","Text":"which is m, and that\u0027s a contradiction."},{"Start":"03:26.095 ","End":"03:30.430","Text":"That\u0027s part c. Part d is the same intervals."},{"Start":"03:30.430 ","End":"03:34.385","Text":"We have to show that b is the supremum."},{"Start":"03:34.385 ","End":"03:37.215","Text":"Certainly b is an upper bound,"},{"Start":"03:37.215 ","End":"03:41.275","Text":"because anything in these intervals is less than b."},{"Start":"03:41.275 ","End":"03:43.810","Text":"We have to show that it\u0027s the least upper bound,"},{"Start":"03:43.810 ","End":"03:45.220","Text":"not just an upper bound."},{"Start":"03:45.220 ","End":"03:47.785","Text":"By contradiction, suppose that\u0027s something smaller,"},{"Start":"03:47.785 ","End":"03:51.250","Text":"call it s, that\u0027s the least upper bound of I."},{"Start":"03:51.250 ","End":"03:54.505","Text":"Now, choose any x in I,"},{"Start":"03:54.505 ","End":"04:00.865","Text":"then x is less than or equal to s. Clear that s is in the interval i,"},{"Start":"04:00.865 ","End":"04:03.940","Text":"it\u0027s equal to x or further right than the x,"},{"Start":"04:03.940 ","End":"04:05.575","Text":"but still left of b,"},{"Start":"04:05.575 ","End":"04:07.870","Text":"take anything in the interval, move further right,"},{"Start":"04:07.870 ","End":"04:09.070","Text":"but still less than b,"},{"Start":"04:09.070 ","End":"04:11.365","Text":"it\u0027s going to still be in the interval."},{"Start":"04:11.365 ","End":"04:15.520","Text":"We have s less than b and we can use the density property of"},{"Start":"04:15.520 ","End":"04:19.300","Text":"the real numbers to find something between s and b,"},{"Start":"04:19.300 ","End":"04:25.785","Text":"call it t. Then t is in the interval because it\u0027s still less than b,"},{"Start":"04:25.785 ","End":"04:27.880","Text":"and t is bigger than s,"},{"Start":"04:27.880 ","End":"04:30.310","Text":"which is the supremum of the interval."},{"Start":"04:30.310 ","End":"04:34.815","Text":"You can\u0027t have something be in the set and be bigger than the upper bound,"},{"Start":"04:34.815 ","End":"04:36.510","Text":"so that\u0027s a contradiction."},{"Start":"04:36.510 ","End":"04:41.005","Text":"The contradiction came from assuming that b is not the least, so it is."},{"Start":"04:41.005 ","End":"04:45.475","Text":"It\u0027s the least upper bound and that\u0027s part d. Now, part e,"},{"Start":"04:45.475 ","End":"04:48.590","Text":"where the interval is 1 of these forms,"},{"Start":"04:48.590 ","End":"04:52.670","Text":"where it\u0027s unbounded on the right or it goes up to infinity."},{"Start":"04:52.670 ","End":"04:55.895","Text":"We have to show that it doesn\u0027t have a supremum,"},{"Start":"04:55.895 ","End":"04:58.430","Text":"whichever 1 of these 3 it is."},{"Start":"04:58.430 ","End":"05:00.130","Text":"We\u0027ll do a proof by contradiction."},{"Start":"05:00.130 ","End":"05:02.765","Text":"Suppose that the interval does have a supremum,"},{"Start":"05:02.765 ","End":"05:05.480","Text":"call that supremum s. Now,"},{"Start":"05:05.480 ","End":"05:07.429","Text":"choose any x in the interval,"},{"Start":"05:07.429 ","End":"05:11.300","Text":"then x is less than or equal to the supremum,"},{"Start":"05:11.300 ","End":"05:14.420","Text":"and any number s is less than s plus 1."},{"Start":"05:14.420 ","End":"05:15.680","Text":"Distinguish cases."},{"Start":"05:15.680 ","End":"05:18.920","Text":"It\u0027s either this case or these 2 cases."},{"Start":"05:18.920 ","End":"05:20.660","Text":"Take this case first."},{"Start":"05:20.660 ","End":"05:22.130","Text":"If this is the interval,"},{"Start":"05:22.130 ","End":"05:23.660","Text":"it\u0027s all of the real numbers."},{"Start":"05:23.660 ","End":"05:26.800","Text":"Certainly s plus 1 is in the interval."},{"Start":"05:26.800 ","End":"05:29.355","Text":"If it\u0027s 1 of the 2 other cases,"},{"Start":"05:29.355 ","End":"05:34.370","Text":"then I claim that s plus 1 is still in the interval because a less than or equal to x,"},{"Start":"05:34.370 ","End":"05:35.630","Text":"x is less than or equal to s,"},{"Start":"05:35.630 ","End":"05:37.430","Text":"s is less than s plus 1."},{"Start":"05:37.430 ","End":"05:40.760","Text":"So s plus 1 is to the right of a."},{"Start":"05:40.760 ","End":"05:45.345","Text":"In these 2 cases, anything to the right of a is still in the interval."},{"Start":"05:45.345 ","End":"05:49.125","Text":"S plus 1 belongs to I in both cases,"},{"Start":"05:49.125 ","End":"05:52.835","Text":"which in the interval is less than or equal to the supremum of the interval,"},{"Start":"05:52.835 ","End":"05:55.820","Text":"but that\u0027s s. We have s plus 1 less than or"},{"Start":"05:55.820 ","End":"05:59.135","Text":"equal to s. That can\u0027t be, that\u0027s a contradiction."},{"Start":"05:59.135 ","End":"06:01.040","Text":"That concludes part e,"},{"Start":"06:01.040 ","End":"06:02.240","Text":"but we\u0027re not quite done yet."},{"Start":"06:02.240 ","End":"06:06.390","Text":"I\u0027m just going to make a remark that in parts b to e,"},{"Start":"06:06.390 ","End":"06:10.820","Text":"you could have corresponding claim instead of supremum and maximum,"},{"Start":"06:10.820 ","End":"06:18.045","Text":"you could change maximum to minimum and supremum to infimum and get 4 similar claims."},{"Start":"06:18.045 ","End":"06:20.170","Text":"They would be proved similarly."},{"Start":"06:20.170 ","End":"06:25.175","Text":"I leave it for you to read them but it exactly mimics the proof for this."},{"Start":"06:25.175 ","End":"06:29.160","Text":"We\u0027re not going to prove them from scratch. We\u0027re done."}],"ID":26581},{"Watched":false,"Name":"Exercise 5","Duration":"2m 36s","ChapterTopicVideoID":25778,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.560","Text":"In this exercise, A is a non-empty set of real numbers and x is some real number."},{"Start":"00:07.560 ","End":"00:14.940","Text":"We define the distance from x to A from a point to a set by d of x and A,"},{"Start":"00:14.940 ","End":"00:20.850","Text":"that\u0027s the notation, is the infimum of the absolute value of x minus A,"},{"Start":"00:20.850 ","End":"00:24.450","Text":"where a runs over all the members of A."},{"Start":"00:24.450 ","End":"00:29.565","Text":"In other words, it\u0027s the infimum distance from x to any point in A."},{"Start":"00:29.565 ","End":"00:35.430","Text":"Suppose Alpha is the least upper bound or supremum of A,"},{"Start":"00:35.430 ","End":"00:41.265","Text":"we have to show that the distance from Alpha to the set A is 0."},{"Start":"00:41.265 ","End":"00:43.205","Text":"If you want to imagine an example,"},{"Start":"00:43.205 ","End":"00:49.820","Text":"take A as the negative numbers and then the least upper bound Alpha is 0."},{"Start":"00:49.820 ","End":"00:52.159","Text":"0 is not in the negative numbers,"},{"Start":"00:52.159 ","End":"00:54.920","Text":"but its distance to the negative numbers is still"},{"Start":"00:54.920 ","End":"00:58.550","Text":"0 because it\u0027s smaller than any positive number."},{"Start":"00:58.550 ","End":"01:00.380","Text":"We\u0027ll prove it by contradiction,"},{"Start":"01:00.380 ","End":"01:07.290","Text":"and suppose that the distance from Alpha to A is positive and choose positive Epsilon,"},{"Start":"01:07.290 ","End":"01:10.470","Text":"which is less than this distance."},{"Start":"01:10.470 ","End":"01:13.770","Text":"Now, I claim that for any a in A,"},{"Start":"01:13.770 ","End":"01:18.555","Text":"the absolute value of Alpha minus a is bigger than Epsilon. Why is that?"},{"Start":"01:18.555 ","End":"01:21.950","Text":"Because Alpha minus a in absolute value is bigger"},{"Start":"01:21.950 ","End":"01:25.640","Text":"than the infimum of absolute value of x minus a,"},{"Start":"01:25.640 ","End":"01:30.495","Text":"and this is bigger than Epsilon because this is our d,"},{"Start":"01:30.495 ","End":"01:32.250","Text":"which is bigger than Epsilon."},{"Start":"01:32.250 ","End":"01:36.650","Text":"This is bigger or equal to something which is bigger than Epsilon."},{"Start":"01:36.650 ","End":"01:41.705","Text":"We can drop the absolute value because Alpha is bigger or equal to a."},{"Start":"01:41.705 ","End":"01:48.305","Text":"The reason for that is that Alpha is the least upper bound of the set A,"},{"Start":"01:48.305 ","End":"01:52.375","Text":"so any element of A is less than or equal to Alpha."},{"Start":"01:52.375 ","End":"01:56.210","Text":"We can change the places of a and Epsilon here,"},{"Start":"01:56.210 ","End":"02:01.845","Text":"and we get Alpha minus Epsilon is bigger than a for all a in A."},{"Start":"02:01.845 ","End":"02:07.345","Text":"This means that Alpha minus Epsilon is an upper bound for the set A,"},{"Start":"02:07.345 ","End":"02:11.395","Text":"but it\u0027s an upper bound which is less than the least upper bound,"},{"Start":"02:11.395 ","End":"02:15.470","Text":"which means that\u0027s a contradiction because we can\u0027t have less than the least."},{"Start":"02:15.470 ","End":"02:19.460","Text":"The contradiction came from assuming this is bigger than 0,"},{"Start":"02:19.460 ","End":"02:22.550","Text":"so it has to be less than or equal to 0,"},{"Start":"02:22.550 ","End":"02:24.365","Text":"but this is not negative."},{"Start":"02:24.365 ","End":"02:28.490","Text":"The distance is the infimum of non-negative quantity,"},{"Start":"02:28.490 ","End":"02:30.140","Text":"so it is also non-negative,"},{"Start":"02:30.140 ","End":"02:32.090","Text":"so can\u0027t be positive,"},{"Start":"02:32.090 ","End":"02:34.045","Text":"can\u0027t be negative, so it\u0027s 0."},{"Start":"02:34.045 ","End":"02:37.350","Text":"That\u0027s what we had to show, so we\u0027re done."}],"ID":26582},{"Watched":false,"Name":"Exercise 6 - N is unbounded","Duration":"1m 6s","ChapterTopicVideoID":25779,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this exercise, we\u0027re going prove something that"},{"Start":"00:02.940 ","End":"00:05.955","Text":"seems obvious but still needs to be proved,"},{"Start":"00:05.955 ","End":"00:09.855","Text":"that the natural numbers are unbounded from above,"},{"Start":"00:09.855 ","End":"00:14.460","Text":"meaning that there\u0027s no real number that\u0027s larger than all the natural numbers."},{"Start":"00:14.460 ","End":"00:16.200","Text":"Like I said, it seems obvious,"},{"Start":"00:16.200 ","End":"00:17.595","Text":"but let\u0027s prove it."},{"Start":"00:17.595 ","End":"00:19.260","Text":"We\u0027ll do it by contradiction,"},{"Start":"00:19.260 ","End":"00:23.190","Text":"and suppose on the contrary that N is bounded from above."},{"Start":"00:23.190 ","End":"00:27.600","Text":"Then we\u0027ll apply the completeness axiom it is non-empty and bounded from above,"},{"Start":"00:27.600 ","End":"00:29.310","Text":"so it has a least upper bound,"},{"Start":"00:29.310 ","End":"00:33.420","Text":"and we\u0027ll call that number S. If S is the least upper bound,"},{"Start":"00:33.420 ","End":"00:36.300","Text":"then S minus 1 is no longer an upper bound."},{"Start":"00:36.300 ","End":"00:37.860","Text":"If this is not an upper bound,"},{"Start":"00:37.860 ","End":"00:42.210","Text":"at least one of the members of N is bigger than it."},{"Start":"00:42.210 ","End":"00:45.705","Text":"Let\u0027s say N is bigger than S minus 1."},{"Start":"00:45.705 ","End":"00:47.515","Text":"Add 1 to both sides,"},{"Start":"00:47.515 ","End":"00:49.580","Text":"and we get that N plus 1 is bigger than"},{"Start":"00:49.580 ","End":"00:53.870","Text":"S. N plus 1 is also a natural number, I should have said."},{"Start":"00:53.870 ","End":"00:56.990","Text":"That contradicts the fact that S is the supremum of"},{"Start":"00:56.990 ","End":"01:00.405","Text":"the natural numbers because here we have one that\u0027s bigger than the supremum."},{"Start":"01:00.405 ","End":"01:06.690","Text":"So this contradiction proves that N is bounded from above. We\u0027re done."}],"ID":26583},{"Watched":false,"Name":"Exercise 7","Duration":"5m 31s","ChapterTopicVideoID":25780,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.249","Text":"This exercise demonstrates 1 of the big differences between the rationals and the reals."},{"Start":"00:05.249 ","End":"00:10.080","Text":"We can find a non-empty subset of the rationals which has an upper bound,"},{"Start":"00:10.080 ","End":"00:12.195","Text":"but doesn\u0027t have a least upper bound,"},{"Start":"00:12.195 ","End":"00:15.465","Text":"and this is in contrast with the reals. We\u0027re given a hint."},{"Start":"00:15.465 ","End":"00:17.609","Text":"What set to try."},{"Start":"00:17.609 ","End":"00:19.890","Text":"There is a short way of solving this."},{"Start":"00:19.890 ","End":"00:22.200","Text":"If you know the properties of the real numbers,"},{"Start":"00:22.200 ","End":"00:24.360","Text":"we\u0027re going to assume that we don\u0027t know about"},{"Start":"00:24.360 ","End":"00:27.060","Text":"the reals or at least we\u0027re not going to use them in this exercise,"},{"Start":"00:27.060 ","End":"00:29.125","Text":"we just know about rational numbers."},{"Start":"00:29.125 ","End":"00:32.990","Text":"Let\u0027s begin. First thing is that we have to show is that"},{"Start":"00:32.990 ","End":"00:36.830","Text":"this set is bounded from above, has an upper-bound."},{"Start":"00:36.830 ","End":"00:38.915","Text":"I\u0027ll give you an example of an upper bound,"},{"Start":"00:38.915 ","End":"00:42.410","Text":"1 1/2 could also use 10 as an upper bound."},{"Start":"00:42.410 ","End":"00:43.910","Text":"Let me demonstrate this."},{"Start":"00:43.910 ","End":"00:45.980","Text":"Suppose r is a rational number."},{"Start":"00:45.980 ","End":"00:47.480","Text":"I should have said r is in Q,"},{"Start":"00:47.480 ","End":"00:49.900","Text":"which is bigger than 1 1/2,"},{"Start":"00:49.900 ","End":"00:52.635","Text":"then r squared is bigger than 2 1/4,"},{"Start":"00:52.635 ","End":"00:54.590","Text":"which is bigger or equal to 2."},{"Start":"00:54.590 ","End":"01:01.690","Text":"So r is not in K. This proves the contrapositive that if r is in K,"},{"Start":"01:01.690 ","End":"01:05.185","Text":"then r is less than or equal to 1 1/2."},{"Start":"01:05.185 ","End":"01:08.545","Text":"K, of course is non-negative because it contains zero, for example."},{"Start":"01:08.545 ","End":"01:11.870","Text":"Now, we have to show that it doesn\u0027t have a least upper bound."},{"Start":"01:11.870 ","End":"01:13.369","Text":"Again, by contradiction,"},{"Start":"01:13.369 ","End":"01:17.420","Text":"suppose that it does have a least upper bound and call that p,"},{"Start":"01:17.420 ","End":"01:18.890","Text":"which is, again, in Q."},{"Start":"01:18.890 ","End":"01:22.550","Text":"We only know about Q. p is positive."},{"Start":"01:22.550 ","End":"01:24.875","Text":"That\u0027s fairly straightforward."},{"Start":"01:24.875 ","End":"01:30.210","Text":"For example, 1 belongs to K because 1 squared is less than 2,"},{"Start":"01:30.210 ","End":"01:32.410","Text":"so p is at least 1."},{"Start":"01:32.410 ","End":"01:34.685","Text":"I need for later that p is positive."},{"Start":"01:34.685 ","End":"01:37.280","Text":"p squared is not equal to 2 because it\u0027s"},{"Start":"01:37.280 ","End":"01:42.230","Text":"a famous proof that there\u0027s no rational number squared which is equal to 2."},{"Start":"01:42.230 ","End":"01:44.120","Text":"If it\u0027s not equal to 2,"},{"Start":"01:44.120 ","End":"01:47.840","Text":"then has to either be less than 2 or bigger than 2,"},{"Start":"01:47.840 ","End":"01:52.585","Text":"and will separate into 2 cases and in each case will get a contradiction."},{"Start":"01:52.585 ","End":"01:56.375","Text":"Case 1, p squared is less than 2."},{"Start":"01:56.375 ","End":"01:59.375","Text":"Now, I\u0027m going to pull an expression out of a hat."},{"Start":"01:59.375 ","End":"02:00.995","Text":"This was done by trial and error."},{"Start":"02:00.995 ","End":"02:05.780","Text":"Let q be equal to 2p plus 2 over p plus 2."},{"Start":"02:05.780 ","End":"02:08.060","Text":"Certainly, q is a rational number,"},{"Start":"02:08.060 ","End":"02:09.680","Text":"because if p is rational,"},{"Start":"02:09.680 ","End":"02:12.350","Text":"all these operations of addition and division and"},{"Start":"02:12.350 ","End":"02:15.985","Text":"multiplication stay within the rational numbers."},{"Start":"02:15.985 ","End":"02:22.655","Text":"I claim that p is less than q and q squared is less than 2."},{"Start":"02:22.655 ","End":"02:26.075","Text":"We\u0027ll show that in a moment. But suppose that we\u0027ve shown this."},{"Start":"02:26.075 ","End":"02:29.455","Text":"This will show that q is in K,"},{"Start":"02:29.455 ","End":"02:32.510","Text":"because q squared is less than 2 and it\u0027s irrational,"},{"Start":"02:32.510 ","End":"02:35.930","Text":"and q is bigger than the supremum of K,"},{"Start":"02:35.930 ","End":"02:38.165","Text":"and that will be a contradiction."},{"Start":"02:38.165 ","End":"02:40.775","Text":"All we have to prove now"},{"Start":"02:40.775 ","End":"02:44.660","Text":"these 2 claims that p is less than q and q squared is less than 2."},{"Start":"02:44.660 ","End":"02:46.490","Text":"So q minus p,"},{"Start":"02:46.490 ","End":"02:48.680","Text":"which is this expression,"},{"Start":"02:48.680 ","End":"02:50.570","Text":"is equal to, and you know what?"},{"Start":"02:50.570 ","End":"02:54.505","Text":"I\u0027m just going to go quickly through this is just algebra."},{"Start":"02:54.505 ","End":"02:58.815","Text":"My simple algebra show that q minus p is bigger than zero."},{"Start":"02:58.815 ","End":"03:01.400","Text":"Here I\u0027ve used the fact that p is positive,"},{"Start":"03:01.400 ","End":"03:03.080","Text":"so p plus 2 is positive,"},{"Start":"03:03.080 ","End":"03:05.330","Text":"so we can divide by a positive number."},{"Start":"03:05.330 ","End":"03:08.080","Text":"Anyway, take your time to pause and follow this."},{"Start":"03:08.080 ","End":"03:12.155","Text":"We conclude that q is bigger than p. Similarly,"},{"Start":"03:12.155 ","End":"03:14.615","Text":"going to show that q squared is less than 2."},{"Start":"03:14.615 ","End":"03:15.980","Text":"We\u0027ll just do that quickly."},{"Start":"03:15.980 ","End":"03:20.365","Text":"In fact, I\u0027ll just expose it all and let you study that."},{"Start":"03:20.365 ","End":"03:23.085","Text":"It\u0027s just very straightforward algebra,"},{"Start":"03:23.085 ","End":"03:25.250","Text":"and we conclude that q squared is less than 2."},{"Start":"03:25.250 ","End":"03:27.065","Text":"So we prove these 2 things,"},{"Start":"03:27.065 ","End":"03:30.315","Text":"and so in this case, we get a contradiction."},{"Start":"03:30.315 ","End":"03:32.300","Text":"Let\u0027s turn to Case 2 now,"},{"Start":"03:32.300 ","End":"03:34.535","Text":"where p squared is bigger than 2."},{"Start":"03:34.535 ","End":"03:39.740","Text":"Again, I\u0027m going to let q equal to p plus 2 over p plus 2 as above,"},{"Start":"03:39.740 ","End":"03:42.640","Text":"q is still a rational number."},{"Start":"03:42.640 ","End":"03:49.895","Text":"The claim is this time that q is less than p and q squared is bigger than 2."},{"Start":"03:49.895 ","End":"03:52.055","Text":"Once we\u0027ve shown this,"},{"Start":"03:52.055 ","End":"03:53.915","Text":"we\u0027ll get to the conclusion we want,"},{"Start":"03:53.915 ","End":"03:57.200","Text":"because we see that q is an upper bound of K. Well,"},{"Start":"03:57.200 ","End":"03:58.685","Text":"maybe I need to explain."},{"Start":"03:58.685 ","End":"04:06.260","Text":"Because q squared is bigger than 2 and never think in K has a square less than 2,"},{"Start":"04:06.260 ","End":"04:08.490","Text":"then q is bigger than anything in K,"},{"Start":"04:08.490 ","End":"04:10.205","Text":"so it\u0027s an upper bound."},{"Start":"04:10.205 ","End":"04:14.105","Text":"It\u0027s less than the supremum because the supremum is p."},{"Start":"04:14.105 ","End":"04:18.440","Text":"So we have an upper bound that\u0027s smaller than the smallest upper bound,"},{"Start":"04:18.440 ","End":"04:20.525","Text":"and that\u0027s a contradiction."},{"Start":"04:20.525 ","End":"04:25.135","Text":"We still have to prove these 2 things and then we\u0027re done."},{"Start":"04:25.135 ","End":"04:27.030","Text":"That\u0027s in the first case."},{"Start":"04:27.030 ","End":"04:28.880","Text":"I\u0027ll go over some of the algebra."},{"Start":"04:28.880 ","End":"04:31.280","Text":"It really is straightforward algebra from this,"},{"Start":"04:31.280 ","End":"04:32.540","Text":"we put a common denominator,"},{"Start":"04:32.540 ","End":"04:35.960","Text":"we get to this, simplifies to this."},{"Start":"04:35.960 ","End":"04:38.240","Text":"We mentioned that p is positive,"},{"Start":"04:38.240 ","End":"04:41.960","Text":"so the denominator is positive and p squared is bigger than 2."},{"Start":"04:41.960 ","End":"04:43.550","Text":"The numerator is positive,"},{"Start":"04:43.550 ","End":"04:45.020","Text":"so this is positive."},{"Start":"04:45.020 ","End":"04:47.030","Text":"If p minus q is positive,"},{"Start":"04:47.030 ","End":"04:52.925","Text":"then p is bigger than q We also have to show that q squared is bigger than 2."},{"Start":"04:52.925 ","End":"04:55.050","Text":"So q squared is this."},{"Start":"04:55.050 ","End":"05:00.465","Text":"Expand that further or if we break this up into 2p squared plus 2p squared."},{"Start":"05:00.465 ","End":"05:02.385","Text":"Now, p squared is bigger than 2."},{"Start":"05:02.385 ","End":"05:06.405","Text":"I\u0027m going to replace this term by 2 times 2,"},{"Start":"05:06.405 ","End":"05:10.350","Text":"which is 4, which added to this 4 gives us the 8 here."},{"Start":"05:10.350 ","End":"05:13.085","Text":"Then this, we can take 2 out the brackets."},{"Start":"05:13.085 ","End":"05:14.780","Text":"This is a perfect square,"},{"Start":"05:14.780 ","End":"05:16.895","Text":"which is p plus 2 squared."},{"Start":"05:16.895 ","End":"05:19.759","Text":"Then we cancel numerator and denominator,"},{"Start":"05:19.759 ","End":"05:23.150","Text":"this factor, so it comes out to be just 2."},{"Start":"05:23.150 ","End":"05:25.630","Text":"Then we have that q squared is bigger than 2."},{"Start":"05:25.630 ","End":"05:28.205","Text":"We\u0027ve shown the 2 things we wanted to show."},{"Start":"05:28.205 ","End":"05:30.155","Text":"That\u0027s all we had left to prove,"},{"Start":"05:30.155 ","End":"05:32.670","Text":"so we are done."}],"ID":26584},{"Watched":false,"Name":"Exercise 8","Duration":"3m 20s","ChapterTopicVideoID":25781,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"The purpose of this exercise is part b,"},{"Start":"00:04.080 ","End":"00:07.110","Text":"which is a variation on the completeness axiom."},{"Start":"00:07.110 ","End":"00:08.715","Text":"What we had until now,"},{"Start":"00:08.715 ","End":"00:11.730","Text":"was that any non-empty set of real numbers which is"},{"Start":"00:11.730 ","End":"00:15.225","Text":"bounded from above has a least upper bound."},{"Start":"00:15.225 ","End":"00:16.529","Text":"This is the equivalent."},{"Start":"00:16.529 ","End":"00:17.730","Text":"If it\u0027s bounded from below,"},{"Start":"00:17.730 ","End":"00:19.710","Text":"it has a greatest lower bound."},{"Start":"00:19.710 ","End":"00:22.575","Text":"Part a, is the preparation for part b."},{"Start":"00:22.575 ","End":"00:27.570","Text":"Part a says if we have a set that\u0027s bounded below by m,"},{"Start":"00:27.570 ","End":"00:34.810","Text":"then minus the set is bounded above by minus m and vice versa."},{"Start":"00:34.810 ","End":"00:39.080","Text":"For example, if we take the set K to be 1 and a half,"},{"Start":"00:39.080 ","End":"00:41.345","Text":"1 and a third, 1 and a quarter, and so on."},{"Start":"00:41.345 ","End":"00:44.410","Text":"It\u0027s bounded below by 1."},{"Start":"00:44.410 ","End":"00:47.265","Text":"If we take minus K,"},{"Start":"00:47.265 ","End":"00:48.840","Text":"which is minus 1 and a half,"},{"Start":"00:48.840 ","End":"00:50.055","Text":"minus 1 and a third,"},{"Start":"00:50.055 ","End":"00:52.085","Text":"minus 1 and a quarter, and so on."},{"Start":"00:52.085 ","End":"00:56.315","Text":"It\u0027s bounded above by minus of what was here."},{"Start":"00:56.315 ","End":"00:58.885","Text":"Let\u0027s talk proving part a."},{"Start":"00:58.885 ","End":"01:01.190","Text":"Let\u0027s say we\u0027re given set K,"},{"Start":"01:01.190 ","End":"01:05.750","Text":"which is bounded below by little m. What we\u0027re going to"},{"Start":"01:05.750 ","End":"01:11.955","Text":"do is to show that minus K is bounded from above by minus m. Let\u0027s do that."},{"Start":"01:11.955 ","End":"01:14.535","Text":"If x is in minus K,"},{"Start":"01:14.535 ","End":"01:21.965","Text":"then minus x is in K. Minus x is bigger or equal to m,"},{"Start":"01:21.965 ","End":"01:25.715","Text":"because m is the lower bound for K."},{"Start":"01:25.715 ","End":"01:29.510","Text":"Then multiplying both sides of the inequality by minus 1,"},{"Start":"01:29.510 ","End":"01:34.630","Text":"we get that x is less than or equal to minus m. That\u0027s how we get from here to here."},{"Start":"01:34.630 ","End":"01:40.555","Text":"Similarly, if we are given that K is bounded from above by m,"},{"Start":"01:40.555 ","End":"01:43.960","Text":"we\u0027re going to show that minus K is bounded from below by"},{"Start":"01:43.960 ","End":"01:48.455","Text":"minus m. Suppose x is in minus K,"},{"Start":"01:48.455 ","End":"01:53.550","Text":"then minus x is in K."},{"Start":"01:53.550 ","End":"01:59.530","Text":"Minus x is less than or equal to m multiplied by minus 1."},{"Start":"01:59.530 ","End":"02:02.065","Text":"X is bigger or equal to minus m,"},{"Start":"02:02.065 ","End":"02:03.715","Text":"which gives us this."},{"Start":"02:03.715 ","End":"02:06.005","Text":"Now on to part b."},{"Start":"02:06.005 ","End":"02:13.960","Text":"Let K be bounded from below by little m. We have to show that K has a greatest lower"},{"Start":"02:13.960 ","End":"02:22.970","Text":"bound minus K is bounded above by minus m. Using the least upper bound property,"},{"Start":"02:22.970 ","End":"02:25.610","Text":"minus K has a least upper bound,"},{"Start":"02:25.610 ","End":"02:28.760","Text":"call it big M. In particular,"},{"Start":"02:28.760 ","End":"02:30.875","Text":"this is an upper bound."},{"Start":"02:30.875 ","End":"02:36.830","Text":"K, which is minus minus K is bounded from below by minus m."},{"Start":"02:36.830 ","End":"02:42.560","Text":"What remains to show is that this minus m is the greatest lower bound."},{"Start":"02:42.560 ","End":"02:44.585","Text":"Suppose is not the greatest,"},{"Start":"02:44.585 ","End":"02:50.810","Text":"let m be another lower bound of K. Then minus m is an upper bound of"},{"Start":"02:50.810 ","End":"02:55.490","Text":"minus K. But big M is the least upper"},{"Start":"02:55.490 ","End":"03:01.430","Text":"bound for minus K. Minus m is bigger or equal to M. Now,"},{"Start":"03:01.430 ","End":"03:04.700","Text":"reversing this inequality or multiplying by minus 1,"},{"Start":"03:04.700 ","End":"03:10.910","Text":"we have m is less than or equal to minus M. That shows that minus M is"},{"Start":"03:10.910 ","End":"03:17.600","Text":"the greatest lower bound because in other lower bound is less than or equal to it."},{"Start":"03:17.600 ","End":"03:21.420","Text":"That concludes this exercise."}],"ID":26585},{"Watched":false,"Name":"Exercise 9","Duration":"3m 25s","ChapterTopicVideoID":25782,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.205","Text":"In this exercise,"},{"Start":"00:02.205 ","End":"00:06.120","Text":"T is the set of real numbers that\u0027s bounded from above"},{"Start":"00:06.120 ","End":"00:10.140","Text":"and S is a subset which is nonempty."},{"Start":"00:10.140 ","End":"00:13.980","Text":"We have to prove 4 things: that T has a supremum,"},{"Start":"00:13.980 ","End":"00:16.170","Text":"that S has a supremum,"},{"Start":"00:16.170 ","End":"00:20.030","Text":"that the supremum of S is less than or equal to the supremum of T,"},{"Start":"00:20.030 ","End":"00:24.390","Text":"and if maximum of S and maximum of T both exist,"},{"Start":"00:24.390 ","End":"00:30.120","Text":"then the maximum of S is less than or equal to the maximum of T. We\u0027ll start with part a."},{"Start":"00:30.120 ","End":"00:34.815","Text":"Note that since S is nonempty and T includes S,"},{"Start":"00:34.815 ","End":"00:36.615","Text":"that T is nonempty."},{"Start":"00:36.615 ","End":"00:42.165","Text":"Though that in more detail S is not empty so there is some element in S,"},{"Start":"00:42.165 ","End":"00:46.650","Text":"but since we have a subset relationship that element is also in T,"},{"Start":"00:46.650 ","End":"00:49.410","Text":"and because T has some element it\u0027s nonempty."},{"Start":"00:49.410 ","End":"00:53.450","Text":"Now, since T is nonempty and bounded from above,"},{"Start":"00:53.450 ","End":"00:56.280","Text":"which is given, it has a supremum."},{"Start":"00:56.280 ","End":"00:59.295","Text":"This is by the completeness axiom."},{"Start":"00:59.295 ","End":"01:04.870","Text":"Part b, we have to show that set S has a supremum."},{"Start":"01:04.870 ","End":"01:11.450","Text":"Let\u0027s take a general element S in S. Because of the subset relation between S and T,"},{"Start":"01:11.450 ","End":"01:17.460","Text":"S is also in T and so S is less than or equal to the supremum of T. S is"},{"Start":"01:17.460 ","End":"01:23.795","Text":"a nonempty set and it\u0027s bounded from above by sup T and by the axiom,"},{"Start":"01:23.795 ","End":"01:25.920","Text":"S has a supremum,"},{"Start":"01:25.920 ","End":"01:28.110","Text":"sup S. That\u0027s part b."},{"Start":"01:28.110 ","End":"01:31.850","Text":"Now on to part c where we have to show this inequality between"},{"Start":"01:31.850 ","End":"01:34.790","Text":"sup S and sup T. Suppose on the contrary"},{"Start":"01:34.790 ","End":"01:37.940","Text":"that the supremum of S was bigger than the supremum of T."},{"Start":"01:37.940 ","End":"01:40.790","Text":"Sup T can\u0027t be an upper bound of"},{"Start":"01:40.790 ","End":"01:43.490","Text":"S because it\u0027s less than the least upper bound"},{"Start":"01:43.490 ","End":"01:46.925","Text":"so it can no longer be an upper bound otherwise we\u0027d have a smaller 1."},{"Start":"01:46.925 ","End":"01:51.750","Text":"There exists some S in S such that S is bigger than sup T,"},{"Start":"01:51.750 ","End":"01:56.540","Text":"but S also belongs to T because S is a subset of T and because S is in T,"},{"Start":"01:56.540 ","End":"01:59.780","Text":"S is less than or equal to supremum of T. This and"},{"Start":"01:59.780 ","End":"02:04.400","Text":"this contradict each other and that contradiction came from this."},{"Start":"02:04.400 ","End":"02:10.250","Text":"That\u0027s part c. We just have part d left to show that if S and T both have a maximum,"},{"Start":"02:10.250 ","End":"02:13.480","Text":"and the maximum of S is less than or equal to the maximum of T,"},{"Start":"02:13.480 ","End":"02:15.330","Text":"that\u0027s fairly straightforward,"},{"Start":"02:15.330 ","End":"02:18.980","Text":"because when a set has a maximum then it\u0027s also the supremum."},{"Start":"02:18.980 ","End":"02:20.870","Text":"They both exist and are equal."},{"Start":"02:20.870 ","End":"02:23.960","Text":"Maximum of S is the same as supremum of S,"},{"Start":"02:23.960 ","End":"02:27.575","Text":"and then by the above this is less than or equal to this,"},{"Start":"02:27.575 ","End":"02:29.455","Text":"and this is equal to this."},{"Start":"02:29.455 ","End":"02:34.475","Text":"Maximum of S is less than or equal to the maximum of T and we\u0027re done."},{"Start":"02:34.475 ","End":"02:36.304","Text":"Wait, I almost forgot."},{"Start":"02:36.304 ","End":"02:38.030","Text":"There is a remark."},{"Start":"02:38.030 ","End":"02:43.025","Text":"This exercise was all about setting a subset and supremum and maximum,"},{"Start":"02:43.025 ","End":"02:48.680","Text":"but there is an entirely analogous version with infimum and minimum."},{"Start":"02:48.680 ","End":"02:52.640","Text":"Let\u0027s suppose that T is bounded from below this time"},{"Start":"02:52.640 ","End":"02:55.895","Text":"and S is a subset which is nonempty,"},{"Start":"02:55.895 ","End":"03:01.170","Text":"then we can prove 4 things: that T has an infimum without a supremum,"},{"Start":"03:01.170 ","End":"03:04.300","Text":"that S has an infimum without a supremum,"},{"Start":"03:04.300 ","End":"03:06.920","Text":"that this is bigger or equal to this"},{"Start":"03:06.920 ","End":"03:11.030","Text":"as opposed to the supremum of the inequality within the other direction,"},{"Start":"03:11.030 ","End":"03:14.545","Text":"and if they both have minimum instead of maximum,"},{"Start":"03:14.545 ","End":"03:17.135","Text":"then we have an inequality this way."},{"Start":"03:17.135 ","End":"03:21.350","Text":"This is entirely analogous to what we showed and proof is almost the same"},{"Start":"03:21.350 ","End":"03:25.650","Text":"so just adding this as a remark. Now we\u0027re done."}],"ID":26586},{"Watched":false,"Name":"Exercise 10","Duration":"1m 39s","ChapterTopicVideoID":25783,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.955","Text":"In this exercise, A and B are sets of real numbers that are non-empty and bounded,"},{"Start":"00:05.955 ","End":"00:12.015","Text":"and we suppose that the supremum of A is equal to the infimum of B."},{"Start":"00:12.015 ","End":"00:14.700","Text":"What we have to prove is that for any Delta,"},{"Start":"00:14.700 ","End":"00:16.525","Text":"which is bigger than 0,"},{"Start":"00:16.525 ","End":"00:22.775","Text":"there exists an x in A and a y in B such that x plus Delta is bigger than y."},{"Start":"00:22.775 ","End":"00:24.050","Text":"These 2 are equal,"},{"Start":"00:24.050 ","End":"00:26.945","Text":"so we can call their common value Alpha."},{"Start":"00:26.945 ","End":"00:29.735","Text":"Alpha is the greatest lower bound of B."},{"Start":"00:29.735 ","End":"00:31.895","Text":"That\u0027s what infimum means."},{"Start":"00:31.895 ","End":"00:34.955","Text":"So if I add Delta to Alpha,"},{"Start":"00:34.955 ","End":"00:40.054","Text":"it can no longer be a lower bound of B because it\u0027s greater than the greatest."},{"Start":"00:40.054 ","End":"00:43.100","Text":"That means that there is some y in B,"},{"Start":"00:43.100 ","End":"00:45.980","Text":"which is less than Alpha plus Delta,"},{"Start":"00:45.980 ","End":"00:49.010","Text":"and from this, we just bring Delta over to the other side,"},{"Start":"00:49.010 ","End":"00:52.825","Text":"we get that y minus Delta is less than Alpha."},{"Start":"00:52.825 ","End":"00:55.230","Text":"Now Alpha is the least upper bound of A,"},{"Start":"00:55.230 ","End":"00:57.875","Text":"and y minus Delta is less than that."},{"Start":"00:57.875 ","End":"01:00.500","Text":"So it can\u0027t be an upper bound of A,"},{"Start":"01:00.500 ","End":"01:02.345","Text":"otherwise, it would be smaller than the lease."},{"Start":"01:02.345 ","End":"01:04.445","Text":"If it\u0027s not an upper bound of A,"},{"Start":"01:04.445 ","End":"01:09.500","Text":"there must be some x in A which is bigger than this y minus Delta."},{"Start":"01:09.500 ","End":"01:11.180","Text":"If it was no such x,"},{"Start":"01:11.180 ","End":"01:15.335","Text":"then it\u0027s less than or equal to y minus Delta for all such x."},{"Start":"01:15.335 ","End":"01:18.170","Text":"So y minus Delta would then be an upper bound."},{"Start":"01:18.170 ","End":"01:20.930","Text":"Now note that x plus Delta is bigger than"},{"Start":"01:20.930 ","End":"01:25.490","Text":"y just by bringing the Delta over to the other side here."},{"Start":"01:25.490 ","End":"01:29.435","Text":"Let\u0027s collect together what I\u0027ve highlighted in color,"},{"Start":"01:29.435 ","End":"01:30.965","Text":"y belongs to B,"},{"Start":"01:30.965 ","End":"01:32.240","Text":"x belongs to A,"},{"Start":"01:32.240 ","End":"01:34.925","Text":"and x plus Delta is bigger than y."},{"Start":"01:34.925 ","End":"01:37.175","Text":"This is exactly what we had to find,"},{"Start":"01:37.175 ","End":"01:39.930","Text":"and so we are done."}],"ID":26587},{"Watched":false,"Name":"Exercise 11","Duration":"1m 42s","ChapterTopicVideoID":25784,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.005","Text":"In this exercise, A and B are 2 sets of real numbers that are non-empty and bounded."},{"Start":"00:07.005 ","End":"00:12.720","Text":"We suppose that the supremum of A is less than or equal to the infimum of B."},{"Start":"00:12.720 ","End":"00:17.010","Text":"We also suppose that for any positive delta,"},{"Start":"00:17.010 ","End":"00:21.150","Text":"there exists a pair x in A and y in B,"},{"Start":"00:21.150 ","End":"00:24.390","Text":"such that x plus delta is bigger than y."},{"Start":"00:24.390 ","End":"00:29.740","Text":"From this, we have to prove that the supremum of A is equal to the infimum of B,"},{"Start":"00:29.740 ","End":"00:32.265","Text":"and we\u0027ll do this by contradiction."},{"Start":"00:32.265 ","End":"00:36.635","Text":"Suppose this is not true because we have a less than or equal to."},{"Start":"00:36.635 ","End":"00:42.500","Text":"If it\u0027s not equal, then the supremum of A will be strictly less than the infimum of B."},{"Start":"00:42.500 ","End":"00:46.235","Text":"We let delta be the difference this minus this,"},{"Start":"00:46.235 ","End":"00:48.370","Text":"then delta is positive."},{"Start":"00:48.370 ","End":"00:53.029","Text":"Now, by the given for any delta that\u0027s positive,"},{"Start":"00:53.029 ","End":"00:55.115","Text":"we have this pair x and y,"},{"Start":"00:55.115 ","End":"00:59.080","Text":"such that x plus delta is bigger than y from here."},{"Start":"00:59.080 ","End":"01:01.115","Text":"What we get is that x,"},{"Start":"01:01.115 ","End":"01:05.405","Text":"which is less than or equal to the supremum of A is equal to."},{"Start":"01:05.405 ","End":"01:08.165","Text":"Just bring this to this side and the delta to the other side."},{"Start":"01:08.165 ","End":"01:13.645","Text":"We have infimum of B minus delta and infimum of B is less than or equal to y."},{"Start":"01:13.645 ","End":"01:18.290","Text":"Look, we have x is less than or equal to y minus delta."},{"Start":"01:18.290 ","End":"01:20.705","Text":"So bring the delta with the other side,"},{"Start":"01:20.705 ","End":"01:23.630","Text":"x plus delta is less than or equal to y."},{"Start":"01:23.630 ","End":"01:25.730","Text":"Now, look at these 2."},{"Start":"01:25.730 ","End":"01:27.910","Text":"I\u0027d say this is a contradiction."},{"Start":"01:27.910 ","End":"01:32.360","Text":"This contradiction came from supposing that this was less"},{"Start":"01:32.360 ","End":"01:36.545","Text":"than this and so that\u0027s not true and so they are equal."},{"Start":"01:36.545 ","End":"01:38.480","Text":"Just like we said, this is less than or equal to,"},{"Start":"01:38.480 ","End":"01:42.780","Text":"but not less than so it\u0027s equal to. Okay, that\u0027s it."}],"ID":26588},{"Watched":false,"Name":"Exercise 12","Duration":"1m 16s","ChapterTopicVideoID":25785,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, the set A is non-empty and"},{"Start":"00:03.510 ","End":"00:07.140","Text":"bounded from above and doesn\u0027t have a maximum."},{"Start":"00:07.140 ","End":"00:10.215","Text":"Let x be less than the supremum of A."},{"Start":"00:10.215 ","End":"00:12.795","Text":"We have to prove that at least 2 members of A"},{"Start":"00:12.795 ","End":"00:17.010","Text":"lie between x and supremum of A."},{"Start":"00:17.010 ","End":"00:19.950","Text":"Now x is less than supremum,"},{"Start":"00:19.950 ","End":"00:22.125","Text":"which is the least upper bound of A,"},{"Start":"00:22.125 ","End":"00:27.330","Text":"so x can\u0027t be an upper bound of A meaning it can\u0027t be less than the least."},{"Start":"00:27.330 ","End":"00:33.525","Text":"There is some a in A such that a is bigger than x."},{"Start":"00:33.525 ","End":"00:37.405","Text":"If all the a were less than or equal to x then x would be an upper bound."},{"Start":"00:37.405 ","End":"00:42.439","Text":"Now, a is the maximum of A because we\u0027re told that set A has no maximum."},{"Start":"00:42.439 ","End":"00:46.610","Text":"There exists some b and A such that b is bigger than a,"},{"Start":"00:46.610 ","End":"00:48.334","Text":"otherwise it would be the maximum."},{"Start":"00:48.334 ","End":"00:50.690","Text":"Similarly, b isn\u0027t the maximum of A,"},{"Start":"00:50.690 ","End":"00:54.365","Text":"so there\u0027s some c in A such that c is bigger than b."},{"Start":"00:54.365 ","End":"00:57.290","Text":"Now what we have is that x is less than a, less than b,"},{"Start":"00:57.290 ","End":"01:01.085","Text":"less than c, which is less than or equal to the supremum."},{"Start":"01:01.085 ","End":"01:06.425","Text":"I just throw out the c and what we get is the following,"},{"Start":"01:06.425 ","End":"01:11.005","Text":"which means that a and b are 2 members of set A,"},{"Start":"01:11.005 ","End":"01:17.420","Text":"which are between x and sup A as required. We\u0027re done."}],"ID":26589},{"Watched":false,"Name":"Exercise 13","Duration":"4m 35s","ChapterTopicVideoID":25786,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.840","Text":"In this exercise, S is a nonempty set of real numbers which is bounded from above,"},{"Start":"00:06.840 ","End":"00:09.795","Text":"and c is a non-negative real number."},{"Start":"00:09.795 ","End":"00:13.350","Text":"We have to prove that the set cS,"},{"Start":"00:13.350 ","End":"00:14.950","Text":"which is defined as follows,"},{"Start":"00:14.950 ","End":"00:18.300","Text":"is nonempty and bounded from above."},{"Start":"00:18.300 ","End":"00:23.460","Text":"That sup of cS is c times sup of S. Just to remark,"},{"Start":"00:23.460 ","End":"00:24.750","Text":"this makes sense,"},{"Start":"00:24.750 ","End":"00:29.570","Text":"these 2 sups exist because S is nonempty and bounded from above,"},{"Start":"00:29.570 ","End":"00:33.175","Text":"and we\u0027re going to prove that cS is nonempty and bounded from above."},{"Start":"00:33.175 ","End":"00:36.135","Text":"Therefore, these 2 supremum exist."},{"Start":"00:36.135 ","End":"00:37.905","Text":"First, the nonempty part,"},{"Start":"00:37.905 ","End":"00:40.050","Text":"since S is nonempty,"},{"Start":"00:40.050 ","End":"00:44.730","Text":"I claim that cS is also nonempty because there is some element x,"},{"Start":"00:44.730 ","End":"00:48.630","Text":"in essence, is nonempty and cx belongs to cS,"},{"Start":"00:48.630 ","End":"00:50.320","Text":"so that\u0027s also nonempty."},{"Start":"00:50.320 ","End":"00:52.460","Text":"Now, we\u0027re going to split into 2 cases."},{"Start":"00:52.460 ","End":"00:53.880","Text":"c is bigger or equal to 0,"},{"Start":"00:53.880 ","End":"00:57.305","Text":"we\u0027ll take equals 0 first and then bigger than 0."},{"Start":"00:57.305 ","End":"00:59.150","Text":"If c equals 0,"},{"Start":"00:59.150 ","End":"01:02.085","Text":"then cS is 0S,"},{"Start":"01:02.085 ","End":"01:05.175","Text":"which is the set of all 0x for x in S,"},{"Start":"01:05.175 ","End":"01:06.470","Text":"but all of these are 0,"},{"Start":"01:06.470 ","End":"01:10.625","Text":"so it\u0027s a singleton set consisting of just 0."},{"Start":"01:10.625 ","End":"01:15.080","Text":"Supremum of this set is the supremum of the singleton 0,"},{"Start":"01:15.080 ","End":"01:16.790","Text":"which is just the 0,"},{"Start":"01:16.790 ","End":"01:20.000","Text":"which also happens to be 0 times the supremum of"},{"Start":"01:20.000 ","End":"01:23.935","Text":"S. Wherever the supremum of S is times 0, it\u0027s still 0."},{"Start":"01:23.935 ","End":"01:28.895","Text":"I should have remarked that because cS is a singleton set,"},{"Start":"01:28.895 ","End":"01:31.790","Text":"is nonempty and bounded from above,"},{"Start":"01:31.790 ","End":"01:33.680","Text":"finite set is always bounded."},{"Start":"01:33.680 ","End":"01:36.170","Text":"That answers this part, nonempty and down from above,"},{"Start":"01:36.170 ","End":"01:38.255","Text":"and we have also proved the equality."},{"Start":"01:38.255 ","End":"01:43.460","Text":"I want to illustrate this with an example but where c is bigger than 0."},{"Start":"01:43.460 ","End":"01:47.090","Text":"Suppose S is the unit interval from 0 to 1,"},{"Start":"01:47.090 ","End":"01:48.770","Text":"and C is 4."},{"Start":"01:48.770 ","End":"01:52.300","Text":"What we have is that the supremum of S is 1."},{"Start":"01:52.300 ","End":"01:54.060","Text":"If we take 4S,"},{"Start":"01:54.060 ","End":"01:56.065","Text":"that\u0027s the interval from 0 to 4."},{"Start":"01:56.065 ","End":"01:59.680","Text":"Take all the xs in here and multiply them by 4,"},{"Start":"01:59.680 ","End":"02:01.905","Text":"we get the integral from 0 to 4."},{"Start":"02:01.905 ","End":"02:07.230","Text":"The supremum of 4S is just the 4 and 4 times 1 is 4,"},{"Start":"02:07.230 ","End":"02:10.535","Text":"so that this formula does hold."},{"Start":"02:10.535 ","End":"02:12.335","Text":"This is just like an example."},{"Start":"02:12.335 ","End":"02:14.930","Text":"Let\u0027s get on to case 2 in general."},{"Start":"02:14.930 ","End":"02:17.285","Text":"So c is bigger than 0,"},{"Start":"02:17.285 ","End":"02:20.420","Text":"and you have to show that it\u0027s bounded from above."},{"Start":"02:20.420 ","End":"02:22.685","Text":"We\u0027ve already shown the nonempty."},{"Start":"02:22.685 ","End":"02:27.380","Text":"The claim is that c times supremum of S is an upper bound of cS."},{"Start":"02:27.380 ","End":"02:28.415","Text":"Let\u0027s prove it."},{"Start":"02:28.415 ","End":"02:31.420","Text":"Suppose y belongs to cS,"},{"Start":"02:31.420 ","End":"02:34.790","Text":"then y is c times some x where x is in"},{"Start":"02:34.790 ","End":"02:38.600","Text":"S and x is less than or equal to the supremum of S,"},{"Start":"02:38.600 ","End":"02:41.240","Text":"of course, and c is bigger than 0,"},{"Start":"02:41.240 ","End":"02:44.645","Text":"so multiply both sides of an inequality by a positive number."},{"Start":"02:44.645 ","End":"02:50.405","Text":"Then we have cx is less than c supremum of S and this is why."},{"Start":"02:50.405 ","End":"02:54.260","Text":"This is what we wanted to show here."},{"Start":"02:54.260 ","End":"02:56.180","Text":"It is bounded above,"},{"Start":"02:56.180 ","End":"03:01.590","Text":"and we even have an upper bound which is c sup S. Just wrote that out."},{"Start":"03:01.590 ","End":"03:04.760","Text":"We know that the supremum is less than or equal"},{"Start":"03:04.760 ","End":"03:08.300","Text":"to any upper bound because the supremum is the least upper bound."},{"Start":"03:08.300 ","End":"03:10.385","Text":"We have this inequality."},{"Start":"03:10.385 ","End":"03:13.500","Text":"Now, let T equal cS."},{"Start":"03:13.500 ","End":"03:16.910","Text":"We can get that S is c minus 1T."},{"Start":"03:16.910 ","End":"03:20.615","Text":"If we get from here to here by multiplying by c,"},{"Start":"03:20.615 ","End":"03:25.010","Text":"we can get original S back by taking all the elements in here and dividing them by c."},{"Start":"03:25.010 ","End":"03:28.100","Text":"C minus 1 is positive"},{"Start":"03:28.100 ","End":"03:31.775","Text":"because the reciprocal of a positive number is still a positive number."},{"Start":"03:31.775 ","End":"03:33.290","Text":"By what we said above,"},{"Start":"03:33.290 ","End":"03:35.435","Text":"but with c minus 1 instead,"},{"Start":"03:35.435 ","End":"03:39.080","Text":"we have that the supremum of S is less than reciprocal of c times"},{"Start":"03:39.080 ","End":"03:42.620","Text":"supremum of T. Multiply both sides by c,"},{"Start":"03:42.620 ","End":"03:45.620","Text":"we have c sup S less than or equal to sup of T."},{"Start":"03:45.620 ","End":"03:50.445","Text":"But this is equal to sup of cS because T is cS."},{"Start":"03:50.445 ","End":"03:52.050","Text":"I want to highlight what\u0027s important."},{"Start":"03:52.050 ","End":"03:59.245","Text":"We have this, and we have that this is less than or equal to this."},{"Start":"03:59.245 ","End":"04:02.000","Text":"That means that we have equality here."},{"Start":"04:02.000 ","End":"04:04.340","Text":"If this is less than or equal to this and vice versa,"},{"Start":"04:04.340 ","End":"04:06.140","Text":"then we have equality."},{"Start":"04:06.140 ","End":"04:09.935","Text":"Now, remark, we dealt with supremum."},{"Start":"04:09.935 ","End":"04:11.660","Text":"We could have an equivalent,"},{"Start":"04:11.660 ","End":"04:14.645","Text":"analogous claim with infimum."},{"Start":"04:14.645 ","End":"04:18.900","Text":"Suppose S is nonempty and bounded from below."},{"Start":"04:18.900 ","End":"04:21.160","Text":"Again, c is non-negative."},{"Start":"04:21.160 ","End":"04:23.610","Text":"Then cS is bounded from below,"},{"Start":"04:23.610 ","End":"04:30.020","Text":"and infimum of cS is c times infimum of S. We\u0027re replacing supremum with infimum,"},{"Start":"04:30.020 ","End":"04:33.305","Text":"and it\u0027s still hold with bounded from below."},{"Start":"04:33.305 ","End":"04:35.910","Text":"That\u0027s it for this clip."}],"ID":26590},{"Watched":false,"Name":"Exercise 14","Duration":"1m 32s","ChapterTopicVideoID":25787,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.470","Text":"In this exercise, A is the set of real numbers which is nonempty"},{"Start":"00:04.470 ","End":"00:08.670","Text":"and bounded from above by some alpha."},{"Start":"00:08.670 ","End":"00:12.630","Text":"Now suppose that for each natural number n,"},{"Start":"00:12.630 ","End":"00:18.690","Text":"there exists an element an of a such that a_n is bigger than"},{"Start":"00:18.690 ","End":"00:25.020","Text":"alpha minus 1 over n. We have to prove that alpha is the supremum of a."},{"Start":"00:25.020 ","End":"00:30.990","Text":"We\u0027ll do this by contradiction by supposing that alpha is not the least upper bound."},{"Start":"00:30.990 ","End":"00:32.775","Text":"We know it is an upper bound."},{"Start":"00:32.775 ","End":"00:36.230","Text":"Let\u0027s suppose that there was a smaller 1, call it beta."},{"Start":"00:36.230 ","End":"00:38.715","Text":"By a previous exercise,"},{"Start":"00:38.715 ","End":"00:40.715","Text":"whenever you have 2 numbers,"},{"Start":"00:40.715 ","End":"00:42.454","Text":"beta smaller than alpha,"},{"Start":"00:42.454 ","End":"00:47.480","Text":"we can find natural number n such that beta is less than alpha"},{"Start":"00:47.480 ","End":"00:52.460","Text":"minus 1 over n. Now we\u0027ll use the given to"},{"Start":"00:52.460 ","End":"00:57.320","Text":"find an a_n which is bigger than this alpha minus 1 over"},{"Start":"00:57.320 ","End":"01:02.540","Text":"n. What we get is that a_n is bigger than alpha minus 1 over n,"},{"Start":"01:02.540 ","End":"01:04.685","Text":"and alpha minus 1 over n is bigger than beta,"},{"Start":"01:04.685 ","End":"01:06.905","Text":"so a_n is bigger than beta,"},{"Start":"01:06.905 ","End":"01:09.470","Text":"and a_n is in a,"},{"Start":"01:09.470 ","End":"01:10.730","Text":"like it says here."},{"Start":"01:10.730 ","End":"01:13.280","Text":"Now we have reached a contradiction because look,"},{"Start":"01:13.280 ","End":"01:17.275","Text":"beta is an upper bound of the set a,"},{"Start":"01:17.275 ","End":"01:18.710","Text":"and if it\u0027s an upper bound,"},{"Start":"01:18.710 ","End":"01:21.985","Text":"we can\u0027t have an element of a that\u0027s bigger than it."},{"Start":"01:21.985 ","End":"01:27.185","Text":"This contradiction came from assuming that alpha is not the least upper bound,"},{"Start":"01:27.185 ","End":"01:29.210","Text":"and the least upper bound is the supremum,"},{"Start":"01:29.210 ","End":"01:32.490","Text":"same thing. We are done."}],"ID":26591},{"Watched":false,"Name":"Exercise 15 - Density of Rationals and Irrationals","Duration":"3m 1s","ChapterTopicVideoID":25788,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"This exercise has 4 parts."},{"Start":"00:02.220 ","End":"00:03.690","Text":"We\u0027ll do 2 parts in this clip,"},{"Start":"00:03.690 ","End":"00:05.025","Text":"and 2 in the next."},{"Start":"00:05.025 ","End":"00:06.540","Text":"The first 2 parts,"},{"Start":"00:06.540 ","End":"00:08.535","Text":"to paraphrase say as follows,"},{"Start":"00:08.535 ","End":"00:11.100","Text":"between any 2 real numbers,"},{"Start":"00:11.100 ","End":"00:12.960","Text":"there\u0027s a rational number,"},{"Start":"00:12.960 ","End":"00:14.430","Text":"and between any 2 real numbers,"},{"Start":"00:14.430 ","End":"00:15.750","Text":"there\u0027s an irrational number,"},{"Start":"00:15.750 ","End":"00:18.045","Text":"and we have to prove both these statements."},{"Start":"00:18.045 ","End":"00:19.680","Text":"Let\u0027s start with the first one,"},{"Start":"00:19.680 ","End":"00:22.845","Text":"that there\u0027s a rational between any 2 real numbers,"},{"Start":"00:22.845 ","End":"00:24.570","Text":"x and y are real numbers,"},{"Start":"00:24.570 ","End":"00:26.085","Text":"x is less than y ."},{"Start":"00:26.085 ","End":"00:28.550","Text":"Let Epsilon be y minus x,"},{"Start":"00:28.550 ","End":"00:30.170","Text":"so Epsilon is positive."},{"Start":"00:30.170 ","End":"00:35.095","Text":"Now, in the previous exercise we showed that we can find a natural number n,"},{"Start":"00:35.095 ","End":"00:38.905","Text":"such that 1 over n is less than Epsilon,"},{"Start":"00:38.905 ","End":"00:40.520","Text":"and of course it\u0027s bigger than 0."},{"Start":"00:40.520 ","End":"00:44.390","Text":"That means that 1 over n is less than y minus x."},{"Start":"00:44.390 ","End":"00:49.444","Text":"Now define a real number c to be this n times x."},{"Start":"00:49.444 ","End":"00:58.550","Text":"We showed that any number has a floor function m such that c is between m and m plus 1,"},{"Start":"00:58.550 ","End":"01:01.685","Text":"bigger or equal to here and strictly less than here."},{"Start":"01:01.685 ","End":"01:05.180","Text":"Now let\u0027s divide this double inequality by n,"},{"Start":"01:05.180 ","End":"01:10.990","Text":"and we get that m over n less than or equal to c over n is x,"},{"Start":"01:10.990 ","End":"01:15.540","Text":"is less than m plus 1 over n. Going to let this be p,"},{"Start":"01:15.540 ","End":"01:20.550","Text":"so let p equal m plus 1 over n. Then x is less than p,"},{"Start":"01:20.550 ","End":"01:22.685","Text":"that\u0027s half of what we had to show."},{"Start":"01:22.685 ","End":"01:26.785","Text":"We still have to show that p is less than y,"},{"Start":"01:26.785 ","End":"01:31.005","Text":"y is equal to y minus x plus x,"},{"Start":"01:31.005 ","End":"01:35.900","Text":"y minus x is bigger than 1 plus n. You see that here,"},{"Start":"01:35.900 ","End":"01:39.755","Text":"and x is bigger or equal to m over n,"},{"Start":"01:39.755 ","End":"01:41.530","Text":"get that from here."},{"Start":"01:41.530 ","End":"01:43.020","Text":"Altogether we have,"},{"Start":"01:43.020 ","End":"01:44.790","Text":"this plus this is m plus 1 over n,"},{"Start":"01:44.790 ","End":"01:52.335","Text":"which is p. So y is bigger than p. That concludes Part a."},{"Start":"01:52.335 ","End":"01:57.500","Text":"Now let\u0027s get onto Part b where we have to find an irrational number between x and y."},{"Start":"01:57.500 ","End":"02:01.100","Text":"Let w be any positive irrational number."},{"Start":"02:01.100 ","End":"02:02.750","Text":"If x is less than y,"},{"Start":"02:02.750 ","End":"02:06.960","Text":"then x over w is less than y over w. Now,"},{"Start":"02:06.960 ","End":"02:08.365","Text":"by Part a,"},{"Start":"02:08.365 ","End":"02:13.000","Text":"there exists a rational number p between x over w,"},{"Start":"02:13.000 ","End":"02:15.870","Text":"and y over w. Between any 2 reals,"},{"Start":"02:15.870 ","End":"02:17.550","Text":"we have a rational number."},{"Start":"02:17.550 ","End":"02:23.490","Text":"Multiply out by w and we\u0027ve got x less than pw less than y."},{"Start":"02:23.490 ","End":"02:26.150","Text":"Now let q equals pw."},{"Start":"02:26.150 ","End":"02:27.955","Text":"This is the q we\u0027re looking for."},{"Start":"02:27.955 ","End":"02:31.250","Text":"You need to explain why q is irrational."},{"Start":"02:31.250 ","End":"02:37.280","Text":"Well, w is p over q and w is irrational and p is rational."},{"Start":"02:37.280 ","End":"02:38.765","Text":"If q were rational,"},{"Start":"02:38.765 ","End":"02:41.780","Text":"then we\u0027d have that rational over rational is irrational,"},{"Start":"02:41.780 ","End":"02:45.170","Text":"which is not possible because rational over rational is rational."},{"Start":"02:45.170 ","End":"02:46.910","Text":"If q was rational,"},{"Start":"02:46.910 ","End":"02:48.305","Text":"we\u0027d get a contradiction,"},{"Start":"02:48.305 ","End":"02:50.255","Text":"so q has to be irrational."},{"Start":"02:50.255 ","End":"02:52.570","Text":"So q is irrational,"},{"Start":"02:52.570 ","End":"02:57.405","Text":"and also q is between x and y because q is pw."},{"Start":"02:57.405 ","End":"03:02.290","Text":"This is what we required, so we\u0027re done."}],"ID":26592},{"Watched":false,"Name":"Exercise 16 - Dense Set, Rational and irrational","Duration":"2m 17s","ChapterTopicVideoID":25789,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"We\u0027ve just completed part a and b,"},{"Start":"00:02.910 ","End":"00:05.970","Text":"and now we\u0027re continuing with c and d. For c,"},{"Start":"00:05.970 ","End":"00:08.040","Text":"we need to know a definition."},{"Start":"00:08.040 ","End":"00:11.880","Text":"A subset S of the real numbers is said to be"},{"Start":"00:11.880 ","End":"00:18.895","Text":"dense if for all real x and for all Epsilon bigger than 0,"},{"Start":"00:18.895 ","End":"00:21.905","Text":"there is some element of S,"},{"Start":"00:21.905 ","End":"00:25.785","Text":"which is in an Epsilon neighborhood of x,"},{"Start":"00:25.785 ","End":"00:29.855","Text":"i.e the absolute value of s minus x is less than Epsilon,"},{"Start":"00:29.855 ","End":"00:32.645","Text":"distance from s to x is less than Epsilon."},{"Start":"00:32.645 ","End":"00:34.385","Text":"Now the exercise."},{"Start":"00:34.385 ","End":"00:41.390","Text":"Given S as a subset of R and has the property that for any 2 elements a and b of R,"},{"Start":"00:41.390 ","End":"00:42.965","Text":"with a less than b,"},{"Start":"00:42.965 ","End":"00:45.590","Text":"there exists an element s in S,"},{"Start":"00:45.590 ","End":"00:48.035","Text":"which is between a and b."},{"Start":"00:48.035 ","End":"00:52.730","Text":"We have to show that if S has this property, then it\u0027s dense."},{"Start":"00:52.730 ","End":"00:56.030","Text":"In part d, we have to show that both,"},{"Start":"00:56.030 ","End":"00:58.895","Text":"the rationals and the irrationals,"},{"Start":"00:58.895 ","End":"01:00.550","Text":"are dense."},{"Start":"01:00.550 ","End":"01:02.955","Text":"Part c first."},{"Start":"01:02.955 ","End":"01:05.555","Text":"Let\u0027s assume that S has this property,"},{"Start":"01:05.555 ","End":"01:08.165","Text":"and we\u0027ll show that S is dense."},{"Start":"01:08.165 ","End":"01:11.880","Text":"Given x and an Epsilon bigger than 0,"},{"Start":"01:11.880 ","End":"01:16.725","Text":"we need to find an s that\u0027s close to x within Epsilon."},{"Start":"01:16.725 ","End":"01:22.160","Text":"Choose a equals x minus Epsilon and b equals x plus Epsilon,"},{"Start":"01:22.160 ","End":"01:24.290","Text":"and apply the property."},{"Start":"01:24.290 ","End":"01:26.465","Text":"There is some s in S,"},{"Start":"01:26.465 ","End":"01:29.230","Text":"which is between a and b."},{"Start":"01:29.230 ","End":"01:32.930","Text":"S is in the interval from x minus Epsilon to x plus"},{"Start":"01:32.930 ","End":"01:36.845","Text":"Epsilon and it\u0027s clear that if s is in this interval,"},{"Start":"01:36.845 ","End":"01:41.300","Text":"then the distance of s to x is less than Epsilon."},{"Start":"01:41.300 ","End":"01:43.505","Text":"That\u0027s part c done."},{"Start":"01:43.505 ","End":"01:45.750","Text":"All we\u0027re left with is part d,"},{"Start":"01:45.750 ","End":"01:47.070","Text":"and there\u0027s 2 parts,"},{"Start":"01:47.070 ","End":"01:49.220","Text":"the rationals and the irrationals,"},{"Start":"01:49.220 ","End":"01:53.030","Text":"and we\u0027ll apply a and b of this exercise."},{"Start":"01:53.030 ","End":"02:00.005","Text":"We showed in part a that Q has this property that between any 2 reals is an element of Q,"},{"Start":"02:00.005 ","End":"02:02.270","Text":"a rational number, and in b,"},{"Start":"02:02.270 ","End":"02:04.190","Text":"we showed that between any 2 a and b,"},{"Start":"02:04.190 ","End":"02:05.935","Text":"there\u0027s an irrational number."},{"Start":"02:05.935 ","End":"02:07.890","Text":"They each have the property,"},{"Start":"02:07.890 ","End":"02:09.660","Text":"and so by part c,"},{"Start":"02:09.660 ","End":"02:11.150","Text":"each of these could be S,"},{"Start":"02:11.150 ","End":"02:13.445","Text":"so each of them is dense."},{"Start":"02:13.445 ","End":"02:15.950","Text":"That proves part d,"},{"Start":"02:15.950 ","End":"02:18.750","Text":"and so we are done."}],"ID":26593},{"Watched":false,"Name":"Exercise 17","Duration":"2m 3s","ChapterTopicVideoID":25790,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.670","Text":"In this exercise, we\u0027re asked to prove that if we have 2 real numbers such"},{"Start":"00:05.670 ","End":"00:11.325","Text":"that the absolute value of their difference is less than 1 over n for all natural n,"},{"Start":"00:11.325 ","End":"00:14.745","Text":"then the 2 numbers are equal to each other."},{"Start":"00:14.745 ","End":"00:16.950","Text":"It seems intuitively true,"},{"Start":"00:16.950 ","End":"00:18.825","Text":"but let\u0027s still prove it."},{"Start":"00:18.825 ","End":"00:22.035","Text":"Let\u0027s call this absolute value of a minus b,"},{"Start":"00:22.035 ","End":"00:23.475","Text":"call it Epsilon,"},{"Start":"00:23.475 ","End":"00:24.810","Text":"and Epsilon, of course,"},{"Start":"00:24.810 ","End":"00:26.850","Text":"is bigger or equal to 0,"},{"Start":"00:26.850 ","End":"00:30.180","Text":"and we\u0027re going to prove that Epsilon equals 0."},{"Start":"00:30.180 ","End":"00:31.890","Text":"If we do that,"},{"Start":"00:31.890 ","End":"00:37.470","Text":"then that will prove our result because we\u0027ll get that absolute value of a minus b is 0,"},{"Start":"00:37.470 ","End":"00:39.285","Text":"and if this is true,"},{"Start":"00:39.285 ","End":"00:42.000","Text":"then a minus b is 0."},{"Start":"00:42.000 ","End":"00:43.740","Text":"I\u0027ll return to this in a moment,"},{"Start":"00:43.740 ","End":"00:47.270","Text":"and if a minus b is 0, then a equals b."},{"Start":"00:47.270 ","End":"00:50.675","Text":"Put an asterisk here because it may not be"},{"Start":"00:50.675 ","End":"00:55.775","Text":"immediately obvious that if the absolute value of something is 0,"},{"Start":"00:55.775 ","End":"00:58.295","Text":"then that something is also 0."},{"Start":"00:58.295 ","End":"01:00.040","Text":"I could prove it."},{"Start":"01:00.040 ","End":"01:02.405","Text":"If x is not 0,"},{"Start":"01:02.405 ","End":"01:04.925","Text":"it\u0027s going to be positive or negative,"},{"Start":"01:04.925 ","End":"01:07.610","Text":"and the absolute value of positive is positive,"},{"Start":"01:07.610 ","End":"01:10.280","Text":"and the absolute value of negative is positive."},{"Start":"01:10.280 ","End":"01:13.220","Text":"Only the absolute value of 0 is 0."},{"Start":"01:13.220 ","End":"01:15.555","Text":"Anyway, that\u0027s fairly clear."},{"Start":"01:15.555 ","End":"01:18.080","Text":"We have to prove that Epsilon is 0, so suppose,"},{"Start":"01:18.080 ","End":"01:20.795","Text":"on the contrary, that Epsilon is not 0."},{"Start":"01:20.795 ","End":"01:23.875","Text":"Because epsilon is bigger or equal to 0,"},{"Start":"01:23.875 ","End":"01:27.175","Text":"that means that Epsilon is bigger than 0,"},{"Start":"01:27.175 ","End":"01:32.255","Text":"and we\u0027ll use the previous exercise where we showed that if we have a positive Epsilon,"},{"Start":"01:32.255 ","End":"01:40.170","Text":"then 1 over n is less than Epsilon for some natural n. This is a contradiction because,"},{"Start":"01:40.170 ","End":"01:41.210","Text":"look, Epsilon,"},{"Start":"01:41.210 ","End":"01:43.250","Text":"which is absolute value of a minus b,"},{"Start":"01:43.250 ","End":"01:45.950","Text":"is bigger than 1 over n. On the other hand,"},{"Start":"01:45.950 ","End":"01:51.870","Text":"we\u0027re given that it\u0027s less than 1 over n for all n, so contradiction."},{"Start":"01:51.870 ","End":"01:57.485","Text":"The contradiction came from assuming that Epsilon is not 0,"},{"Start":"01:57.485 ","End":"01:59.330","Text":"so Epsilon is 0,"},{"Start":"01:59.330 ","End":"02:01.340","Text":"and the rest of it follows,"},{"Start":"02:01.340 ","End":"02:04.380","Text":"like we said. We\u0027re done."}],"ID":26594},{"Watched":false,"Name":"Exercise 18","Duration":"4m 54s","ChapterTopicVideoID":25791,"CourseChapterTopicPlaylistID":246307,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"This exercise is about the nested intervals theorem."},{"Start":"00:04.065 ","End":"00:05.595","Text":"Here\u0027s what it says."},{"Start":"00:05.595 ","End":"00:12.450","Text":"Let I_n be a sequence of intervals of the form I_n is the closed interval a_n,"},{"Start":"00:12.450 ","End":"00:18.315","Text":"b_n and suppose that each I_n plus 1 is a subset of"},{"Start":"00:18.315 ","End":"00:24.555","Text":"I_n for all n. We say that the intervals are nested in this case."},{"Start":"00:24.555 ","End":"00:28.300","Text":"Here\u0027s a picture which might give you an idea."},{"Start":"00:28.300 ","End":"00:36.235","Text":"We have to prove that the intersection of all the intervals I_n is not empty."},{"Start":"00:36.235 ","End":"00:37.840","Text":"Then in part b,"},{"Start":"00:37.840 ","End":"00:41.180","Text":"we define a sequence of open intervals,"},{"Start":"00:41.180 ","End":"00:46.190","Text":"0 to 1 over n. We do this for each natural number n. I"},{"Start":"00:46.190 ","End":"00:51.560","Text":"guess I should have specified n plus the natural numbers excluding the 0."},{"Start":"00:51.560 ","End":"00:55.460","Text":"Otherwise, we can\u0027t have the 1 over n. We have to prove that"},{"Start":"00:55.460 ","End":"00:59.735","Text":"the intersection of these intervals I_n is the empty set."},{"Start":"00:59.735 ","End":"01:04.505","Text":"In part c, we\u0027re asked about what seems to be a contradiction."},{"Start":"01:04.505 ","End":"01:07.700","Text":"The sequence I_n in part b is nested."},{"Start":"01:07.700 ","End":"01:13.415","Text":"Each I_n plus 1 is contained in the previous I_n."},{"Start":"01:13.415 ","End":"01:18.030","Text":"The question is, does the result of b contradict the result of a?"},{"Start":"01:18.030 ","End":"01:21.155","Text":"Let\u0027s start solving it, we\u0027ll start with part a,"},{"Start":"01:21.155 ","End":"01:23.855","Text":"where we have these nested intervals,"},{"Start":"01:23.855 ","End":"01:29.255","Text":"which means that each interval contains the subsequent interval."},{"Start":"01:29.255 ","End":"01:33.550","Text":"To write them out, I_1 is a_1 b_1 and so on."},{"Start":"01:33.550 ","End":"01:36.560","Text":"We can also write the following,"},{"Start":"01:36.560 ","End":"01:39.470","Text":"that a_1 less than or equal to a_2 and so on."},{"Start":"01:39.470 ","End":"01:42.385","Text":"The a_Is are increasing,"},{"Start":"01:42.385 ","End":"01:45.495","Text":"but the b_Is are decreasing."},{"Start":"01:45.495 ","End":"01:47.670","Text":"I mean b_1 is bigger or equal to b_2,"},{"Start":"01:47.670 ","End":"01:49.480","Text":"bigger or equal to b_3."},{"Start":"01:49.480 ","End":"01:56.080","Text":"Now, the claim is that the supremum of the a_n exists, call it Alpha."},{"Start":"01:56.080 ","End":"02:01.640","Text":"Just to remark, we could equally well have used Beta equals the infimum of the b_n here,"},{"Start":"02:01.640 ","End":"02:03.290","Text":"but we have to go with 1 of them."},{"Start":"02:03.290 ","End":"02:06.565","Text":"So we choose the supremum of the a_n."},{"Start":"02:06.565 ","End":"02:12.890","Text":"Note that the set a_n is non-empty and bounded from above."},{"Start":"02:12.890 ","End":"02:16.640","Text":"In fact, any of the b_m bounded from above because all the"},{"Start":"02:16.640 ","End":"02:20.950","Text":"a_n\u0027s are bounded above by each of the b_n."},{"Start":"02:20.950 ","End":"02:23.165","Text":"Non-empty and bounded above,"},{"Start":"02:23.165 ","End":"02:27.590","Text":"we have the completeness axiom or the upper bound axiom."},{"Start":"02:27.590 ","End":"02:30.500","Text":"There is such an Alpha which is the supremum,"},{"Start":"02:30.500 ","End":"02:34.935","Text":"and each of the a_n is less than or equal to Alpha for all n. Now,"},{"Start":"02:34.935 ","End":"02:37.935","Text":"each b_m is an upper bound of the a_n."},{"Start":"02:37.935 ","End":"02:40.460","Text":"So b_m has to be bigger or equal to"},{"Start":"02:40.460 ","End":"02:44.000","Text":"Alpha because Alpha is the least upper bound of the a_n."},{"Start":"02:44.000 ","End":"02:48.010","Text":"The least upper bound is less than or equal to just any upper bound."},{"Start":"02:48.010 ","End":"02:51.470","Text":"Alpha, we already know it\u0027s bigger than or equal to all of"},{"Start":"02:51.470 ","End":"02:54.830","Text":"the a_n and it\u0027s less than or equal to all of the b_n."},{"Start":"02:54.830 ","End":"02:59.540","Text":"So Alpha belongs to the interval a_n b_n, which is I_n."},{"Start":"02:59.540 ","End":"03:05.115","Text":"This is for all n in N. Now, because Alpha is in each of the intervals,"},{"Start":"03:05.115 ","End":"03:08.810","Text":"Alpha belongs to the intersection of the intervals."},{"Start":"03:08.810 ","End":"03:11.150","Text":"If Alpha belongs to the intersection,"},{"Start":"03:11.150 ","End":"03:12.980","Text":"then the intersection is non-empty."},{"Start":"03:12.980 ","End":"03:14.855","Text":"It has at least 1 member,"},{"Start":"03:14.855 ","End":"03:18.555","Text":"which is Alpha. That\u0027s part a."},{"Start":"03:18.555 ","End":"03:20.835","Text":"Now, in part b,"},{"Start":"03:20.835 ","End":"03:24.710","Text":"I_n is the open interval 0 to 1 over n. We have to"},{"Start":"03:24.710 ","End":"03:29.230","Text":"prove that the intersection of these I_n is empty."},{"Start":"03:29.230 ","End":"03:31.820","Text":"We\u0027ll do this by contradiction."},{"Start":"03:31.820 ","End":"03:35.085","Text":"Suppose that it isn\u0027t empty."},{"Start":"03:35.085 ","End":"03:38.915","Text":"There is some x that belongs to this intersection."},{"Start":"03:38.915 ","End":"03:40.850","Text":"Now, x belongs to all of the I_n."},{"Start":"03:40.850 ","End":"03:42.890","Text":"In particular, x belongs to I_1,"},{"Start":"03:42.890 ","End":"03:44.540","Text":"which is 0, 1,"},{"Start":"03:44.540 ","End":"03:46.535","Text":"so x is positive."},{"Start":"03:46.535 ","End":"03:49.145","Text":"Now, by a previous exercise,"},{"Start":"03:49.145 ","End":"03:56.275","Text":"there is a natural number m such that 0 is less than 1 over m, less than x."},{"Start":"03:56.275 ","End":"04:00.740","Text":"This implies that x is not in the interval I_m."},{"Start":"04:00.740 ","End":"04:03.080","Text":"x is not between 0 and 1 over m."},{"Start":"04:03.080 ","End":"04:08.720","Text":"That contradicts the fact that x belongs to the intersection of these,"},{"Start":"04:08.720 ","End":"04:10.940","Text":"so it belongs to each 1 of them."},{"Start":"04:10.940 ","End":"04:15.575","Text":"That contradiction shows that the intersection is empty."},{"Start":"04:15.575 ","End":"04:19.340","Text":"Now, part c, summarizing part a,"},{"Start":"04:19.340 ","End":"04:23.060","Text":"we showed that the intersection of these I_n,"},{"Start":"04:23.060 ","End":"04:25.595","Text":"is not empty and these are nested."},{"Start":"04:25.595 ","End":"04:28.010","Text":"In part b, we showed that the intersection of"},{"Start":"04:28.010 ","End":"04:32.459","Text":"these I_n is empty where I_n is as follows,"},{"Start":"04:32.459 ","End":"04:34.035","Text":"and they\u0027re also nested."},{"Start":"04:34.035 ","End":"04:36.080","Text":"Why is there a problem?"},{"Start":"04:36.080 ","End":"04:39.230","Text":"There\u0027s no contradiction because these are closed,"},{"Start":"04:39.230 ","End":"04:44.390","Text":"whereas these are open and that\u0027s a big difference."},{"Start":"04:44.390 ","End":"04:47.105","Text":"The nested intervals theorem applies to"},{"Start":"04:47.105 ","End":"04:51.890","Text":"closed intervals and can\u0027t be expected to apply to open intervals."},{"Start":"04:51.890 ","End":"04:55.110","Text":"In fact, it doesn\u0027t. Okay. We\u0027re done."}],"ID":26595}],"Thumbnail":null,"ID":246307},{"Name":"Further properties of bounded sets","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Completeness Axiom","Duration":"4m 27s","ChapterTopicVideoID":25813,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this lesson, we\u0027ll learn about the completeness axiom for"},{"Start":"00:04.530 ","End":"00:09.450","Text":"the real numbers and it\u0027s equivalent to something called the least upper bound property."},{"Start":"00:09.450 ","End":"00:13.110","Text":"This answers a question I asked in an earlier lecture."},{"Start":"00:13.110 ","End":"00:15.465","Text":"I\u0027m going back to this clip,"},{"Start":"00:15.465 ","End":"00:18.195","Text":"we asked the following 2 questions."},{"Start":"00:18.195 ","End":"00:22.560","Text":"Does a set which is bounded from above necessarily have a least upper bound?"},{"Start":"00:22.560 ","End":"00:27.015","Text":"There is a set which is bounded from below necessarily have a greatest lower bound."},{"Start":"00:27.015 ","End":"00:29.280","Text":"That is the least upper bound property,"},{"Start":"00:29.280 ","End":"00:32.580","Text":"which is equivalent to the greatest lower bound property."},{"Start":"00:32.580 ","End":"00:36.344","Text":"Let\u0027s start by considering an example, the set C,"},{"Start":"00:36.344 ","End":"00:39.525","Text":"which is a 1/2, 2/3,"},{"Start":"00:39.525 ","End":"00:42.070","Text":"3/4, 4/4, n over n plus 1."},{"Start":"00:42.070 ","End":"00:44.975","Text":"Basically, this set C,"},{"Start":"00:44.975 ","End":"00:48.785","Text":"certainly bounded above, for example, by 1."},{"Start":"00:48.785 ","End":"00:51.020","Text":"All of these are less than 1."},{"Start":"00:51.020 ","End":"00:53.015","Text":"Not the only upper-bound,"},{"Start":"00:53.015 ","End":"00:55.249","Text":"3 is an upper bound also,"},{"Start":"00:55.249 ","End":"00:57.365","Text":"magically anything big or equal to 1."},{"Start":"00:57.365 ","End":"01:01.460","Text":"The question is, does it necessarily have a least upper bound,"},{"Start":"01:01.460 ","End":"01:05.360","Text":"the supremum emphasizing the word necessarily, I mean,"},{"Start":"01:05.360 ","End":"01:07.910","Text":"we could check in this particular case and see"},{"Start":"01:07.910 ","End":"01:10.655","Text":"whether it does or doesn\u0027t have a least upper bound."},{"Start":"01:10.655 ","End":"01:12.095","Text":"But the question really means,"},{"Start":"01:12.095 ","End":"01:16.175","Text":"can I deduce this from the mere fact that it is bounded above?"},{"Start":"01:16.175 ","End":"01:18.260","Text":"It turns out that in this case,"},{"Start":"01:18.260 ","End":"01:20.450","Text":"1 is the least upper bound."},{"Start":"01:20.450 ","End":"01:22.730","Text":"That\u0027s not hard to show that."},{"Start":"01:22.730 ","End":"01:27.890","Text":"My question is, can we always say that every non-empty set which is bounded from above,"},{"Start":"01:27.890 ","End":"01:30.325","Text":"has at least upper bound."},{"Start":"01:30.325 ","End":"01:32.460","Text":"For the real numbers,"},{"Start":"01:32.460 ","End":"01:34.835","Text":"as I said the answer is yes,"},{"Start":"01:34.835 ","End":"01:38.330","Text":"but it turns out that it depends on what set of numbers we\u0027re talking about."},{"Start":"01:38.330 ","End":"01:39.785","Text":"What is the universal set,"},{"Start":"01:39.785 ","End":"01:42.895","Text":"which is usually reals or rationals."},{"Start":"01:42.895 ","End":"01:44.550","Text":"For the case of the rationals,"},{"Start":"01:44.550 ","End":"01:46.020","Text":"the answer is no."},{"Start":"01:46.020 ","End":"01:48.865","Text":"We will give a counter-example below."},{"Start":"01:48.865 ","End":"01:50.390","Text":"For the case of the reals,"},{"Start":"01:50.390 ","End":"01:56.614","Text":"the answer is yes and this is known as the completeness axiom for the real numbers."},{"Start":"01:56.614 ","End":"01:59.270","Text":"We won\u0027t prove this in this course,"},{"Start":"01:59.270 ","End":"02:01.040","Text":"it\u0027s beyond the scope."},{"Start":"02:01.040 ","End":"02:05.510","Text":"What we will do is give the example that this is not true for the rational."},{"Start":"02:05.510 ","End":"02:10.040","Text":"Consider the following set K in the rational numbers,"},{"Start":"02:10.040 ","End":"02:14.975","Text":"set of all r such that r squared is less than 2."},{"Start":"02:14.975 ","End":"02:18.440","Text":"First of all, note that K is bounded above."},{"Start":"02:18.440 ","End":"02:21.320","Text":"For example, buy 1 and 1/2,"},{"Start":"02:21.320 ","End":"02:23.420","Text":"not the only upper bound."},{"Start":"02:23.420 ","End":"02:26.005","Text":"You could take 10 as an upper bound also,"},{"Start":"02:26.005 ","End":"02:28.800","Text":"if r is bigger than 1 and 1/2,"},{"Start":"02:28.800 ","End":"02:32.040","Text":"then r squared is bigger than 2 and 1/4,"},{"Start":"02:32.040 ","End":"02:33.705","Text":"which is bigger or equal to 2."},{"Start":"02:33.705 ","End":"02:38.985","Text":"R does not belong to K because r squared is not less than 2."},{"Start":"02:38.985 ","End":"02:44.660","Text":"The contra-positive of this is that if r does belong to K,"},{"Start":"02:44.660 ","End":"02:47.645","Text":"then r is less than or equal to 1 and 1/2."},{"Start":"02:47.645 ","End":"02:50.240","Text":"1 and 1/2 is an upper bound."},{"Start":"02:50.240 ","End":"02:56.480","Text":"In the exercises, we\u0027ll see that K does not have a least upper bound."},{"Start":"02:56.480 ","End":"03:03.545","Text":"In summary, the mere definition of supremum of a set doesn\u0027t guarantee its existence,"},{"Start":"03:03.545 ","End":"03:06.770","Text":"depends on which number set we\u0027re talking about,"},{"Start":"03:06.770 ","End":"03:09.050","Text":"mean the guaranteed depends."},{"Start":"03:09.050 ","End":"03:12.595","Text":"Now let\u0027s state the completeness axiom"},{"Start":"03:12.595 ","End":"03:16.160","Text":"formally also known as the least upper bound property,"},{"Start":"03:16.160 ","End":"03:19.280","Text":"is that any non-empty set of"},{"Start":"03:19.280 ","End":"03:23.635","Text":"real numbers which is bounded from above has a least upper bound."},{"Start":"03:23.635 ","End":"03:26.015","Text":"I stressed real because as we said,"},{"Start":"03:26.015 ","End":"03:28.675","Text":"does not hold true for the rationals."},{"Start":"03:28.675 ","End":"03:35.270","Text":"Now the least upper bound property is equivalent to the greatest lower bound property."},{"Start":"03:35.270 ","End":"03:36.905","Text":"You might think it\u0027s not symmetrical."},{"Start":"03:36.905 ","End":"03:40.790","Text":"We could have taken the greatest lower bound as the axiom 1 or the other."},{"Start":"03:40.790 ","End":"03:43.200","Text":"Each can be proven from the other 1."},{"Start":"03:43.200 ","End":"03:46.460","Text":"Greatest Lower bound property says that any non-empty set of"},{"Start":"03:46.460 ","End":"03:50.480","Text":"real numbers which is bounded from below has a greatest lower bound."},{"Start":"03:50.480 ","End":"03:53.270","Text":"Proof will be given in the exercises."},{"Start":"03:53.270 ","End":"03:55.820","Text":"I\u0027ll give you the general outline."},{"Start":"03:55.820 ","End":"03:58.865","Text":"If a is bounded above by some m,"},{"Start":"03:58.865 ","End":"04:06.200","Text":"then the set minus a is bounded above by minus m. It\u0027s a simple exercise in inequalities."},{"Start":"04:06.200 ","End":"04:09.365","Text":"Minus a has a least upper bound,"},{"Start":"04:09.365 ","End":"04:11.840","Text":"call it big M. It\u0027s bounded from above,"},{"Start":"04:11.840 ","End":"04:13.100","Text":"so it has at least upper bound."},{"Start":"04:13.100 ","End":"04:17.120","Text":"Then follows again using inequalities that a itself which"},{"Start":"04:17.120 ","End":"04:21.475","Text":"is minus minus a as minus M as a greatest lower bound."},{"Start":"04:21.475 ","End":"04:24.845","Text":"With more detail, we\u0027ll see this in the exercises."},{"Start":"04:24.845 ","End":"04:27.810","Text":"Okay, that\u0027s it for this clip."}],"ID":26617},{"Watched":false,"Name":"Exercise 1","Duration":"2m 54s","ChapterTopicVideoID":25815,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this exercise, A and B are"},{"Start":"00:03.150 ","End":"00:07.425","Text":"both sets of numbers that are nonempty and bounded from above."},{"Start":"00:07.425 ","End":"00:10.590","Text":"In part a, we\u0027re given that for all x in A,"},{"Start":"00:10.590 ","End":"00:15.840","Text":"there exists a y in B such that an x is less than y and from this we have to"},{"Start":"00:15.840 ","End":"00:18.630","Text":"prove that the supremum of A is less than or"},{"Start":"00:18.630 ","End":"00:21.810","Text":"equal to the supremum of B and the question is,"},{"Start":"00:21.810 ","End":"00:27.820","Text":"can we conclude from this that the supremum of A is strictly less than the supremum of B?"},{"Start":"00:27.820 ","End":"00:32.015","Text":"In part b, we suppose in addition to part a,"},{"Start":"00:32.015 ","End":"00:33.845","Text":"that for all y in B,"},{"Start":"00:33.845 ","End":"00:37.805","Text":"there exists an x in A such that y is less than x."},{"Start":"00:37.805 ","End":"00:43.400","Text":"In this case, we have to prove that the supremum of A is equal to the supremum of B."},{"Start":"00:43.400 ","End":"00:48.150","Text":"From the given, which is that these are nonempty and bounded from above,"},{"Start":"00:48.150 ","End":"00:52.535","Text":"then we conclude that supremum of A and supremum of B both exist."},{"Start":"00:52.535 ","End":"00:56.810","Text":"This is by the completeness axiom or the upper bound axiom."},{"Start":"00:56.810 ","End":"01:00.660","Text":"In part a, first thing we\u0027ll do is prove that"},{"Start":"01:00.660 ","End":"01:04.490","Text":"an upper bound u of B is also an upper bound of A."},{"Start":"01:04.490 ","End":"01:05.825","Text":"That\u0027s the first step."},{"Start":"01:05.825 ","End":"01:11.915","Text":"Choose any x in A but given there exists y in B such that x is less than y."},{"Start":"01:11.915 ","End":"01:16.430","Text":"So y is less than or equal to u because u is an upper bound of B."},{"Start":"01:16.430 ","End":"01:19.220","Text":"If x is less than y and y is less than or equal to u,"},{"Start":"01:19.220 ","End":"01:20.975","Text":"then x is less than u."},{"Start":"01:20.975 ","End":"01:23.120","Text":"What we\u0027ve shown is that if x is in A,"},{"Start":"01:23.120 ","End":"01:24.380","Text":"then x is less than u,"},{"Start":"01:24.380 ","End":"01:27.040","Text":"and this means that u is an upper bound of A."},{"Start":"01:27.040 ","End":"01:29.355","Text":"Now we\u0027ll take a very particular u."},{"Start":"01:29.355 ","End":"01:32.640","Text":"We\u0027ll take u as the supremum of B. u is"},{"Start":"01:32.640 ","End":"01:36.080","Text":"an upper bound of B because that\u0027s what supremum of B is,"},{"Start":"01:36.080 ","End":"01:38.270","Text":"and by what we just showed here,"},{"Start":"01:38.270 ","End":"01:40.855","Text":"it\u0027s also an upper bound of A."},{"Start":"01:40.855 ","End":"01:43.760","Text":"Now the supremum of A is the least upper bound,"},{"Start":"01:43.760 ","End":"01:45.690","Text":"so it\u0027s less than or equal to any upper bound,"},{"Start":"01:45.690 ","End":"01:47.330","Text":"so it\u0027s less than or equal to u,"},{"Start":"01:47.330 ","End":"01:49.265","Text":"which happens to be supremum of B."},{"Start":"01:49.265 ","End":"01:52.295","Text":"This shows that sup A is less than or equal to sup B."},{"Start":"01:52.295 ","End":"01:54.080","Text":"That\u0027s what we have to show."},{"Start":"01:54.080 ","End":"01:58.680","Text":"However, we can\u0027t conclude that sup A is strictly"},{"Start":"01:58.680 ","End":"02:03.615","Text":"less than sup B and I\u0027ll give you a counterexample."},{"Start":"02:03.615 ","End":"02:09.440","Text":"Let A and B both be the same, the interval 0,1."},{"Start":"02:09.440 ","End":"02:14.220","Text":"Now they satisfy the condition because any x you give me in A,"},{"Start":"02:14.220 ","End":"02:16.730","Text":"I can find a y in B that\u0027s bigger than it."},{"Start":"02:16.730 ","End":"02:19.640","Text":"I mean, any number between 0 and 1 has another number in 0;"},{"Start":"02:19.640 ","End":"02:20.885","Text":"1 that\u0027s bigger than it."},{"Start":"02:20.885 ","End":"02:24.360","Text":"It satisfies the condition that sup A equals sup"},{"Start":"02:24.360 ","End":"02:28.375","Text":"B because A and B are both the same and it\u0027s equal to 1 in fact."},{"Start":"02:28.375 ","End":"02:31.045","Text":"Yeah, so the answer to that is no."},{"Start":"02:31.045 ","End":"02:32.620","Text":"In part b,"},{"Start":"02:32.620 ","End":"02:35.405","Text":"I\u0027m going to use the technique of part a."},{"Start":"02:35.405 ","End":"02:39.180","Text":"On the 1 hand, we have sup A less than or equal to sup B by"},{"Start":"02:39.180 ","End":"02:42.860","Text":"the given and we also have the opposite condition,"},{"Start":"02:42.860 ","End":"02:46.515","Text":"which gives us that sup B is less than or equal to sup A."},{"Start":"02:46.515 ","End":"02:51.125","Text":"Both these inequalities give us that supremum of A is equal to supremum of B."},{"Start":"02:51.125 ","End":"02:54.480","Text":"That completes part b and we\u0027re done."}],"ID":26619},{"Watched":false,"Name":"Exercise 2","Duration":"4m 6s","ChapterTopicVideoID":25816,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.088","Text":"In this exercise, S and T are both non-empty"},{"Start":"00:04.088 ","End":"00:05.669","Text":"and bounded from above"},{"Start":"00:05.669 ","End":"00:08.670","Text":"and we have to prove that S plus T,"},{"Start":"00:08.670 ","End":"00:11.565","Text":"which is defined as the set of all x plus y,"},{"Start":"00:11.565 ","End":"00:13.710","Text":"where x is in S and y is in T."},{"Start":"00:13.710 ","End":"00:16.755","Text":"That this set is non-empty,"},{"Start":"00:16.755 ","End":"00:22.680","Text":"bounded from above and we have the following equality and all of these to promote"},{"Start":"00:22.680 ","End":"00:25.830","Text":"exist because everything is non-empty and bounded and"},{"Start":"00:25.830 ","End":"00:29.415","Text":"we have to prove it the sup of S plus T is supS plus supT."},{"Start":"00:29.415 ","End":"00:35.520","Text":"We\u0027ll start with pointing out that S plus T is nonempty because S is not empty so"},{"Start":"00:35.520 ","End":"00:36.960","Text":"we can choose an x in S."},{"Start":"00:36.960 ","End":"00:39.660","Text":"T is nonempty so we can choose a y in T."},{"Start":"00:39.660 ","End":"00:42.075","Text":"If we add x plus y,"},{"Start":"00:42.075 ","End":"00:46.595","Text":"by definition it belongs to S plus T so S plus T is not empty."},{"Start":"00:46.595 ","End":"00:50.030","Text":"Now from the given and by the upper bound axiom,"},{"Start":"00:50.030 ","End":"00:53.675","Text":"both sup S and sup T exist."},{"Start":"00:53.675 ","End":"00:56.495","Text":"Our question now is what about S plus T?"},{"Start":"00:56.495 ","End":"00:57.620","Text":"It\u0027s nonempty."},{"Start":"00:57.620 ","End":"01:00.005","Text":"We have to show that it has an upper bound."},{"Start":"01:00.005 ","End":"01:01.475","Text":"It\u0027s bounded from above."},{"Start":"01:01.475 ","End":"01:05.660","Text":"The claim is that supS plus supT is an upper bound for"},{"Start":"01:05.660 ","End":"01:13.865","Text":"the set S plus T. I will show this by showing that if you take any z in S plus T,"},{"Start":"01:13.865 ","End":"01:17.020","Text":"then it\u0027s less than or equal to supS plus supT."},{"Start":"01:17.020 ","End":"01:19.170","Text":"If z is in S plus T,"},{"Start":"01:19.170 ","End":"01:22.505","Text":"then we can write it as x plus y with x in here, y in here."},{"Start":"01:22.505 ","End":"01:28.610","Text":"Since x is in S, it\u0027s less than or equal to sup S. Since y is in T,"},{"Start":"01:28.610 ","End":"01:33.260","Text":"then y is less than or equal to the supremum of T and so z,"},{"Start":"01:33.260 ","End":"01:39.290","Text":"which is x plus y, is less than or equal to supS plus supT as claimed."},{"Start":"01:39.290 ","End":"01:43.280","Text":"Now we have an upper bound for S plus T."},{"Start":"01:43.280 ","End":"01:45.980","Text":"Let\u0027s show us the least upper bound and we\u0027ll"},{"Start":"01:45.980 ","End":"01:52.835","Text":"show this by showing that no upper bound u of S plus T is smaller than supS plus supT,"},{"Start":"01:52.835 ","End":"01:54.340","Text":"meaning it\u0027s the smallest."},{"Start":"01:54.340 ","End":"01:56.420","Text":"We\u0027ll do this by contradiction."},{"Start":"01:56.420 ","End":"01:58.505","Text":"Suppose that there is a u,"},{"Start":"01:58.505 ","End":"02:01.370","Text":"which is an upper bound of S plus T,"},{"Start":"02:01.370 ","End":"02:03.980","Text":"which is smaller than supS plus supT."},{"Start":"02:03.980 ","End":"02:07.850","Text":"Just to remind you what it means to be an upper bound of"},{"Start":"02:07.850 ","End":"02:13.145","Text":"S plus T means that if you take any z in S plus T,"},{"Start":"02:13.145 ","End":"02:15.635","Text":"it\u0027s less than or equal to u."},{"Start":"02:15.635 ","End":"02:18.460","Text":"Now, if this is less than this,"},{"Start":"02:18.460 ","End":"02:21.530","Text":"then the Epsilon be the gap between them."},{"Start":"02:21.530 ","End":"02:25.775","Text":"Other words, it\u0027s going to be supS plus supT minus u,"},{"Start":"02:25.775 ","End":"02:27.680","Text":"so Epsilon is positive."},{"Start":"02:27.680 ","End":"02:34.095","Text":"Now we define S naught to be sup of S minus a half Epsilon."},{"Start":"02:34.095 ","End":"02:36.750","Text":"Since this is smaller than supS,"},{"Start":"02:36.750 ","End":"02:38.910","Text":"there must be some x in S,"},{"Start":"02:38.910 ","End":"02:40.880","Text":"which is bigger than s naught."},{"Start":"02:40.880 ","End":"02:45.095","Text":"Because this is no longer an upper bound because it\u0027s less than the least upper bound."},{"Start":"02:45.095 ","End":"02:48.640","Text":"It\u0027s not an upper bound so some member of the set is bigger than it."},{"Start":"02:48.640 ","End":"02:53.805","Text":"Similarly, we define t naught to be supT minus a half Epsilon"},{"Start":"02:53.805 ","End":"02:59.485","Text":"and there exists a y in T such that y is bigger than t naught."},{"Start":"02:59.485 ","End":"03:04.360","Text":"So x plus y is in S plus T. So x plus y is less than or"},{"Start":"03:04.360 ","End":"03:09.534","Text":"equal to u because u is an upper bound of S plus T. But on the other hand,"},{"Start":"03:09.534 ","End":"03:13.375","Text":"x plus y is bigger than s naught plus t naught."},{"Start":"03:13.375 ","End":"03:15.640","Text":"Because this and this,"},{"Start":"03:15.640 ","End":"03:19.755","Text":"and s naught is this,"},{"Start":"03:19.755 ","End":"03:26.670","Text":"and t naught is this so we get supS plus supT minus Epsilon, which is u."},{"Start":"03:26.670 ","End":"03:29.170","Text":"Now look what we have, on the 1 hand,"},{"Start":"03:29.170 ","End":"03:31.050","Text":"we have this inequality."},{"Start":"03:31.050 ","End":"03:33.920","Text":"On the other hand, we have x plus y is bigger than u,"},{"Start":"03:33.920 ","End":"03:37.040","Text":"so it can\u0027t be less than or equal to and also bigger than so"},{"Start":"03:37.040 ","End":"03:40.675","Text":"that\u0027s a contradiction and that proves our claim."},{"Start":"03:40.675 ","End":"03:43.700","Text":"We\u0027re done except that I want to point something"},{"Start":"03:43.700 ","End":"03:47.450","Text":"out that just as we had with an upper bound and"},{"Start":"03:47.450 ","End":"03:51.305","Text":"supremum is a completely analogous claim"},{"Start":"03:51.305 ","End":"03:54.815","Text":"that if S and T are nonempty and bounded from below,"},{"Start":"03:54.815 ","End":"04:00.830","Text":"then S plus T is bounded from below and inf of the sum is the sum of the inf."},{"Start":"04:00.830 ","End":"04:06.390","Text":"But we\u0027re not going to prove that because it\u0027s completely analogous. Now we\u0027re done."}],"ID":26620},{"Watched":false,"Name":"Exercise 3","Duration":"3m 31s","ChapterTopicVideoID":25817,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.169","Text":"In this exercise, S and T are sets of real numbers"},{"Start":"00:04.169 ","End":"00:07.570","Text":"and they\u0027re both nonempty and bounded from above."},{"Start":"00:07.570 ","End":"00:09.630","Text":"We have to prove, first of all,"},{"Start":"00:09.630 ","End":"00:14.010","Text":"that the union of S and T is nonempty and bounded from above."},{"Start":"00:14.010 ","End":"00:20.399","Text":"Secondly, that the supremum of S union T is the maximum of the 2 suprema."},{"Start":"00:20.399 ","End":"00:23.415","Text":"Like to illustrate with an example before we solve it."},{"Start":"00:23.415 ","End":"00:27.540","Text":"Suppose that S is the integral from 1-2,"},{"Start":"00:27.540 ","End":"00:30.315","Text":"and T is the integral from 1/2 to 4."},{"Start":"00:30.315 ","End":"00:32.475","Text":"The supremum of S is 2,"},{"Start":"00:32.475 ","End":"00:34.500","Text":"supremum of T is 4."},{"Start":"00:34.500 ","End":"00:36.840","Text":"If we take S union T,"},{"Start":"00:36.840 ","End":"00:38.715","Text":"that\u0027s from 1-4,"},{"Start":"00:38.715 ","End":"00:41.505","Text":"and the supremum of this is 4."},{"Start":"00:41.505 ","End":"00:45.795","Text":"Notice that 4 is the maximum of 2 and 4."},{"Start":"00:45.795 ","End":"00:48.075","Text":"It works in this case."},{"Start":"00:48.075 ","End":"00:51.195","Text":"Part a, the nonempty part is easy."},{"Start":"00:51.195 ","End":"00:54.180","Text":"If even 1 of them was nonempty,"},{"Start":"00:54.180 ","End":"00:56.565","Text":"then the union is nonempty."},{"Start":"00:56.565 ","End":"00:57.980","Text":"If both of them are nonempty,"},{"Start":"00:57.980 ","End":"01:01.220","Text":"then certainly as for the bounded from above,"},{"Start":"01:01.220 ","End":"01:05.390","Text":"first of all, S and T are nonempty and bounded from above,"},{"Start":"01:05.390 ","End":"01:08.060","Text":"so sup S and sup T both exist."},{"Start":"01:08.060 ","End":"01:12.350","Text":"The claim is that we take the maximum of these 2 suprema,"},{"Start":"01:12.350 ","End":"01:14.795","Text":"call it M, that\u0027s this here."},{"Start":"01:14.795 ","End":"01:18.290","Text":"This is an upper bound of the set S union T."},{"Start":"01:18.290 ","End":"01:21.650","Text":"M is bigger or equal to the supremum of S"},{"Start":"01:21.650 ","End":"01:26.060","Text":"because the maximum is always bigger or equal to each 1 of the 2 separately."},{"Start":"01:26.060 ","End":"01:27.890","Text":"M is an upper bound of a."},{"Start":"01:27.890 ","End":"01:34.280","Text":"That means that if x is in S and x is less than or equal to M. Similarly, if x is in T,"},{"Start":"01:34.280 ","End":"01:39.380","Text":"then x is less than or equal to M. If x is an S union T, then either way,"},{"Start":"01:39.380 ","End":"01:41.120","Text":"if it\u0027s in S is less than or equal to M,"},{"Start":"01:41.120 ","End":"01:42.860","Text":"and if it\u0027s in T is less than or equal to"},{"Start":"01:42.860 ","End":"01:45.965","Text":"M. X is less than or equal to M for any x in here."},{"Start":"01:45.965 ","End":"01:51.875","Text":"M is an upper bound for S union T. We\u0027ve checked the nonempty part."},{"Start":"01:51.875 ","End":"01:55.400","Text":"Also, it\u0027s bounded from above by this M,"},{"Start":"01:55.400 ","End":"01:57.520","Text":"and so it has a supremum."},{"Start":"01:57.520 ","End":"02:02.900","Text":"What\u0027s more, the supremum of S union T is less than or equal to M,"},{"Start":"02:02.900 ","End":"02:07.340","Text":"because M is an upper bound and this is the least upper bound of S union T."},{"Start":"02:07.340 ","End":"02:10.130","Text":"Now we have to show equality here."},{"Start":"02:10.130 ","End":"02:12.440","Text":"To show equality, we just have to show"},{"Start":"02:12.440 ","End":"02:16.160","Text":"the inequality in the other direction that M is less than or equal to"},{"Start":"02:16.160 ","End":"02:21.560","Text":"the supremum of S union T. We\u0027re going to use a result of a previous exercise which says"},{"Start":"02:21.560 ","End":"02:24.230","Text":"that if we have 2 sets of numbers that are bounded"},{"Start":"02:24.230 ","End":"02:26.960","Text":"from above and 1 is contained in the other,"},{"Start":"02:26.960 ","End":"02:28.220","Text":"and they both have a supremum"},{"Start":"02:28.220 ","End":"02:30.860","Text":"and the supremum of A is less than or equal to the supremum of B."},{"Start":"02:30.860 ","End":"02:32.810","Text":"We\u0027ll use that result in general."},{"Start":"02:32.810 ","End":"02:38.285","Text":"Note that S is contained in S union T and the both nonempty and bounded from above."},{"Start":"02:38.285 ","End":"02:42.100","Text":"Sup S is less than or equal to sup of S union T."},{"Start":"02:42.100 ","End":"02:47.045","Text":"Similarly supremum of T less than or equal to this same thing."},{"Start":"02:47.045 ","End":"02:50.810","Text":"If this is less than this and this is less than the same thing,"},{"Start":"02:50.810 ","End":"02:54.440","Text":"then the maximum of both of them is less than or equal to M."},{"Start":"02:54.440 ","End":"02:58.975","Text":"This proves what we still had to show and so we\u0027re done,"},{"Start":"02:58.975 ","End":"03:00.990","Text":"not quite don\u0027t go just yet."},{"Start":"03:00.990 ","End":"03:02.765","Text":"I still want to make a remark,"},{"Start":"03:02.765 ","End":"03:08.180","Text":"there\u0027s a completely analogous exercise if we change upper bound"},{"Start":"03:08.180 ","End":"03:12.040","Text":"to lower bound and sup to infer, etc."},{"Start":"03:12.040 ","End":"03:13.699","Text":"We can prove the following."},{"Start":"03:13.699 ","End":"03:17.555","Text":"If S and T are nonempty and bounded from below,"},{"Start":"03:17.555 ","End":"03:22.040","Text":"then S union T is nonempty and bounded from below."},{"Start":"03:22.040 ","End":"03:26.210","Text":"This time the inf rather than the sup and the min is [inaudible] of the max,"},{"Start":"03:26.210 ","End":"03:32.370","Text":"the infimum of the union is the minimum of 2 separate infimum. Now we\u0027re done."}],"ID":26621},{"Watched":false,"Name":"Exercise 4","Duration":"2m 25s","ChapterTopicVideoID":25818,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we have 3 sets, S, T, U,"},{"Start":"00:04.170 ","End":"00:07.125","Text":"and they\u0027re all nonempty and bounded from above,"},{"Start":"00:07.125 ","End":"00:09.855","Text":"so we know that they all have a supremum."},{"Start":"00:09.855 ","End":"00:15.780","Text":"Now suppose that for every pair s in S and t in T there exists u in U,"},{"Start":"00:15.780 ","End":"00:19.740","Text":"such that u is bigger or equal to s plus t. We have to"},{"Start":"00:19.740 ","End":"00:24.150","Text":"prove that the supremum of U bigger or equal to supS plus supT."},{"Start":"00:24.150 ","End":"00:29.355","Text":"Like I said, all these 3 suprema exist by the upper bound axiom."},{"Start":"00:29.355 ","End":"00:31.649","Text":"Suppose, on the contrary,"},{"Start":"00:31.649 ","End":"00:32.970","Text":"this is not true,"},{"Start":"00:32.970 ","End":"00:36.270","Text":"that means that this is less than sign,"},{"Start":"00:36.270 ","End":"00:40.585","Text":"supU is less than supS plus supT and we\u0027ll reach a contradiction."},{"Start":"00:40.585 ","End":"00:43.265","Text":"Let Epsilon be the difference."},{"Start":"00:43.265 ","End":"00:44.870","Text":"If this is bigger than this,"},{"Start":"00:44.870 ","End":"00:48.395","Text":"then right-hand side minus left-hand side is a positive Epsilon."},{"Start":"00:48.395 ","End":"00:54.930","Text":"Now define s_0 to be supS minus 1/2 Epsilon,"},{"Start":"00:54.930 ","End":"00:59.450","Text":"then s_0 is less than supS because it\u0027s 1/2 Epsilon less."},{"Start":"00:59.450 ","End":"01:02.720","Text":"It\u0027s no longer an upper bound because it\u0027s less than the least."},{"Start":"01:02.720 ","End":"01:04.570","Text":"There is some x in S,"},{"Start":"01:04.570 ","End":"01:06.780","Text":"such that x is bigger than s_0."},{"Start":"01:06.780 ","End":"01:09.295","Text":"Similarly, we define t_0,"},{"Start":"01:09.295 ","End":"01:15.615","Text":"and we get that there is some y in t such that y is bigger than t_0,"},{"Start":"01:15.615 ","End":"01:17.525","Text":"where t_0 is this."},{"Start":"01:17.525 ","End":"01:22.220","Text":"Now x plus y is bigger than s_0 plus t_0,"},{"Start":"01:22.220 ","End":"01:26.060","Text":"and s_0 is equal to this and t_0 is equal to this,"},{"Start":"01:26.060 ","End":"01:27.545","Text":"so we get this expression."},{"Start":"01:27.545 ","End":"01:32.580","Text":"SupS plus supT minus Epsilon,"},{"Start":"01:32.580 ","End":"01:35.005","Text":"but this is equal to supU."},{"Start":"01:35.005 ","End":"01:37.010","Text":"Now we\u0027ll use the given,"},{"Start":"01:37.010 ","End":"01:42.690","Text":"but with x and y instead of s and t. We have s in S and T is in T,"},{"Start":"01:42.690 ","End":"01:47.720","Text":"so we have here that x is in S and Y is in t. That\u0027s like our S in T,"},{"Start":"01:47.720 ","End":"01:54.680","Text":"so there exists a u which is bigger or equal to x plus y, which is in U."},{"Start":"01:54.680 ","End":"01:56.000","Text":"We have a contradiction."},{"Start":"01:56.000 ","End":"02:03.500","Text":"Look, u is bigger or equal to x plus y and x plus y is bigger than supU."},{"Start":"02:03.500 ","End":"02:07.440","Text":"So u is bigger than supremum of U,"},{"Start":"02:07.440 ","End":"02:11.420","Text":"which contradicts the definition of a supremum because u is in the set U,"},{"Start":"02:11.420 ","End":"02:14.690","Text":"and so has to be less than or equal to the upper bound."},{"Start":"02:14.690 ","End":"02:19.820","Text":"This contradiction proves that this assumption is correct,"},{"Start":"02:19.820 ","End":"02:22.880","Text":"that the assumption of less than is incorrect."},{"Start":"02:22.880 ","End":"02:25.770","Text":"That concludes this exercise."}],"ID":26622},{"Watched":false,"Name":"Exercise 5","Duration":"4m ","ChapterTopicVideoID":25819,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this exercise, S and T are sets of real numbers,"},{"Start":"00:04.575 ","End":"00:06.495","Text":"and they\u0027re both nonempty,"},{"Start":"00:06.495 ","End":"00:12.780","Text":"and we\u0027re given the condition that whenever you have a pair of s in S and t in T,"},{"Start":"00:12.780 ","End":"00:16.170","Text":"then s is less than or equal to t."},{"Start":"00:16.170 ","End":"00:20.280","Text":"We have to prove that the supremum of S and the infimum of T both"},{"Start":"00:20.280 ","End":"00:27.690","Text":"exist and satisfy the inequality sup S less than or equal to inf T. In the second part,"},{"Start":"00:27.690 ","End":"00:29.295","Text":"we have a single set S,"},{"Start":"00:29.295 ","End":"00:31.260","Text":"which is nonempty and bounded,"},{"Start":"00:31.260 ","End":"00:36.630","Text":"and we have to prove that inf S and sup S both"},{"Start":"00:36.630 ","End":"00:43.135","Text":"exist and satisfy the inequality that the infimum is less than the supremum."},{"Start":"00:43.135 ","End":"00:46.895","Text":"The question is, under what condition does equality hold?"},{"Start":"00:46.895 ","End":"00:49.730","Text":"Part a, t is not empty,"},{"Start":"00:49.730 ","End":"00:54.805","Text":"so you can choose t_naught in T. Now by the given,"},{"Start":"00:54.805 ","End":"01:02.730","Text":"s is less than or equal to t_naught for all s in S. This shows that t_naught is"},{"Start":"01:02.730 ","End":"01:06.800","Text":"an upper bound for S. S is bounded from above and"},{"Start":"01:06.800 ","End":"01:12.605","Text":"it\u0027s nonempty by the given and so sup S exists."},{"Start":"01:12.605 ","End":"01:16.250","Text":"We also have that sup S is less than or equal to t_naught"},{"Start":"01:16.250 ","End":"01:20.060","Text":"because the least upper bound is less than or equal to any upper bound."},{"Start":"01:20.060 ","End":"01:25.950","Text":"Now, S is nonempty so we can choose s_naught in S. By the given,"},{"Start":"01:25.950 ","End":"01:29.930","Text":"s_naught is less than or equal to t for all t in T. This time,"},{"Start":"01:29.930 ","End":"01:36.169","Text":"T is bounded from below by s_naught and is nonempty by the given."},{"Start":"01:36.169 ","End":"01:41.355","Text":"That means that infimum of T exists and inf T is bigger or equal to"},{"Start":"01:41.355 ","End":"01:46.909","Text":"s_naught because the greatest lower bound is bigger or equal to any lower bound."},{"Start":"01:46.909 ","End":"01:49.140","Text":"Now, there was nothing special about s_naught,"},{"Start":"01:49.140 ","End":"01:55.130","Text":"we chose any element of big S. Infimum T is"},{"Start":"01:55.130 ","End":"01:58.460","Text":"an upper bound for the set S because"},{"Start":"01:58.460 ","End":"02:01.820","Text":"any member of s is less than or equal to inf T. Now,"},{"Start":"02:01.820 ","End":"02:05.650","Text":"any upper bound is bigger or equal to the least upper bound,"},{"Start":"02:05.650 ","End":"02:11.195","Text":"so inf T is bigger or equal to sup S. That concludes Part a."},{"Start":"02:11.195 ","End":"02:13.130","Text":"Now on to Part b."},{"Start":"02:13.130 ","End":"02:16.670","Text":"Recall that besides the upper bound axiom,"},{"Start":"02:16.670 ","End":"02:20.540","Text":"there\u0027s also a lower bound axiom which is entirely equivalent."},{"Start":"02:20.540 ","End":"02:22.955","Text":"From these and from the given,"},{"Start":"02:22.955 ","End":"02:27.365","Text":"the infimum of S and the supremum of S both exist."},{"Start":"02:27.365 ","End":"02:29.030","Text":"We said that S is bounded,"},{"Start":"02:29.030 ","End":"02:31.945","Text":"which means bounded from below and bounded from above."},{"Start":"02:31.945 ","End":"02:34.920","Text":"We have sup S and inf S,"},{"Start":"02:34.920 ","End":"02:38.025","Text":"and S is nonempty by the given."},{"Start":"02:38.025 ","End":"02:43.310","Text":"Choose any member little s of set S. What we"},{"Start":"02:43.310 ","End":"02:48.740","Text":"have is that s is bigger or equal to the inf of the set S,"},{"Start":"02:48.740 ","End":"02:52.400","Text":"because any member is bigger or equal to the greatest lower bound,"},{"Start":"02:52.400 ","End":"02:56.290","Text":"and it\u0027s also less than or equal to the least upper bound or any upper bound,"},{"Start":"02:56.290 ","End":"02:58.265","Text":"so we have this inequality."},{"Start":"02:58.265 ","End":"03:01.370","Text":"Now, throw out the middleman, get rid of this s,"},{"Start":"03:01.370 ","End":"03:03.770","Text":"so we have inf S less than or equal to sup"},{"Start":"03:03.770 ","End":"03:06.860","Text":"S. That\u0027s what we have to show but there was also the question,"},{"Start":"03:06.860 ","End":"03:08.990","Text":"when does equality hold here?"},{"Start":"03:08.990 ","End":"03:14.365","Text":"Well, suppose that inf S equals sup S, call that a."},{"Start":"03:14.365 ","End":"03:21.370","Text":"In that case, the claim is that S is the singleton set consisting of just a."},{"Start":"03:21.370 ","End":"03:24.935","Text":"Let little s be an element of set S,"},{"Start":"03:24.935 ","End":"03:29.090","Text":"then the infimum of S is less than or equal to S,"},{"Start":"03:29.090 ","End":"03:33.745","Text":"which is less than or equal to sup S. But this is a and this is a,"},{"Start":"03:33.745 ","End":"03:37.190","Text":"so that means that s is equal to a. I"},{"Start":"03:37.190 ","End":"03:40.670","Text":"guess I forgot to say that the converse is also true."},{"Start":"03:40.670 ","End":"03:44.240","Text":"That if S is a singleton set a,"},{"Start":"03:44.240 ","End":"03:49.790","Text":"then obviously the supremum and the infimum of this set are equal and both equal to a."},{"Start":"03:49.790 ","End":"03:53.990","Text":"That is the answer to the part about when does equality hold."},{"Start":"03:53.990 ","End":"03:56.990","Text":"It\u0027s when the set is a singleton set."},{"Start":"03:56.990 ","End":"04:01.470","Text":"Anyway, we\u0027ve concluded Part b also and so we\u0027re done."}],"ID":26623},{"Watched":false,"Name":"The Arcimedean Property","Duration":"3m 3s","ChapterTopicVideoID":25812,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"Now we come to another important property of the real numbers,"},{"Start":"00:03.900 ","End":"00:08.370","Text":"the Archimedean property, also known as the Archimedean principle."},{"Start":"00:08.370 ","End":"00:11.785","Text":"Archimedean axiom, Archimedean theorem,"},{"Start":"00:11.785 ","End":"00:15.915","Text":"they\u0027re all the same. What does it say?"},{"Start":"00:15.915 ","End":"00:18.700","Text":"Let a and b be positive real numbers."},{"Start":"00:18.700 ","End":"00:24.150","Text":"Then there exists a natural number n such that na is bigger than b."},{"Start":"00:24.150 ","End":"00:28.230","Text":"Seems intuitively obvious but it\u0027s still needs proof."},{"Start":"00:28.230 ","End":"00:31.395","Text":"The proof will be in the exercises."},{"Start":"00:31.395 ","End":"00:33.530","Text":"Here\u0027s an example."},{"Start":"00:33.530 ","End":"00:37.745","Text":"Let a be the square root of 2 and b equals 4."},{"Start":"00:37.745 ","End":"00:41.600","Text":"Then you could take n equals 3 or larger,"},{"Start":"00:41.600 ","End":"00:43.790","Text":"and na will be bigger than b."},{"Start":"00:43.790 ","End":"00:48.440","Text":"Let\u0027s see, is 3 times square root of 2 bigger than 4, yes."},{"Start":"00:48.440 ","End":"00:50.375","Text":"Because if you square this,"},{"Start":"00:50.375 ","End":"00:52.070","Text":"we get 3 squared times 2,"},{"Start":"00:52.070 ","End":"00:54.740","Text":"which is 18, which is bigger than 16,"},{"Start":"00:54.740 ","End":"00:55.820","Text":"which is 4 squared."},{"Start":"00:55.820 ","End":"00:57.140","Text":"If this squared is bigger than this squared,"},{"Start":"00:57.140 ","End":"00:59.000","Text":"then this is bigger than this."},{"Start":"00:59.000 ","End":"01:02.105","Text":"Here\u0027s a variant of the Archimedean property."},{"Start":"01:02.105 ","End":"01:04.760","Text":"For any real number x,"},{"Start":"01:04.760 ","End":"01:09.384","Text":"there exists a natural number n such that n is bigger than x."},{"Start":"01:09.384 ","End":"01:12.530","Text":"Proof, it follows from the Archimedean property,"},{"Start":"01:12.530 ","End":"01:20.015","Text":"this version by letting a equals 1 and b equals x. Yeah, here."},{"Start":"01:20.015 ","End":"01:22.490","Text":"If a equals 1 and b equals x,"},{"Start":"01:22.490 ","End":"01:25.520","Text":"we get exactly what it says here."},{"Start":"01:25.520 ","End":"01:28.580","Text":"N times 1 is bigger than x."},{"Start":"01:28.580 ","End":"01:33.095","Text":"A corollary, a conclusion from the Archimedean property,"},{"Start":"01:33.095 ","End":"01:39.790","Text":"is that the set of natural numbers is a subset of the reals is unbounded."},{"Start":"01:39.790 ","End":"01:41.705","Text":"I\u0027ll give 2 proofs."},{"Start":"01:41.705 ","End":"01:44.390","Text":"The first proof by contradiction."},{"Start":"01:44.390 ","End":"01:50.600","Text":"Suppose that some x is an upper bound for n. By the above,"},{"Start":"01:50.600 ","End":"01:52.685","Text":"we can find n,"},{"Start":"01:52.685 ","End":"01:55.414","Text":"natural number that\u0027s bigger than x."},{"Start":"01:55.414 ","End":"02:02.140","Text":"But that means that x is not an upper bound because some member of n is bigger than it."},{"Start":"02:02.140 ","End":"02:06.890","Text":"That\u0027s a contradiction because we said that x is an upper bound."},{"Start":"02:06.890 ","End":"02:09.710","Text":"This contradiction proves our corollary."},{"Start":"02:09.710 ","End":"02:12.095","Text":"Now, here\u0027s an alternative proof."},{"Start":"02:12.095 ","End":"02:15.695","Text":"Again by contradiction, we suppose that x is an upper bound for"},{"Start":"02:15.695 ","End":"02:20.195","Text":"n. This time we\u0027ll use the least upper bound property of the reals."},{"Start":"02:20.195 ","End":"02:23.225","Text":"This set n has an upper bound,"},{"Start":"02:23.225 ","End":"02:25.250","Text":"and therefore it has a least upper bound,"},{"Start":"02:25.250 ","End":"02:28.220","Text":"call it S. Because it\u0027s the least upper bound,"},{"Start":"02:28.220 ","End":"02:31.024","Text":"S minus 1 is not an upper bound."},{"Start":"02:31.024 ","End":"02:34.010","Text":"Otherwise, there\u0027d be a smaller upper bound."},{"Start":"02:34.010 ","End":"02:40.250","Text":"This means that n is bigger than S minus 1 for some natural enemy."},{"Start":"02:40.250 ","End":"02:43.190","Text":"If it\u0027s more than our upper bound then there\u0027s something that\u0027s bigger than it."},{"Start":"02:43.190 ","End":"02:46.850","Text":"Then add 1 to both sides here we\u0027ve got m plus 1 is bigger"},{"Start":"02:46.850 ","End":"02:50.709","Text":"than S. But n plus 1 is also a natural number."},{"Start":"02:50.709 ","End":"02:53.810","Text":"That means that S is not an upper bound of"},{"Start":"02:53.810 ","End":"02:57.155","Text":"the natural numbers and that\u0027s a contradiction."},{"Start":"02:57.155 ","End":"03:01.055","Text":"Once again, the set n is unbounded."},{"Start":"03:01.055 ","End":"03:03.780","Text":"That concludes this clip."}],"ID":26616},{"Watched":false,"Name":"Exercise 6","Duration":"3m 53s","ChapterTopicVideoID":25802,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we\u0027re given a set A,"},{"Start":"00:03.660 ","End":"00:08.280","Text":"which is set of all a_n defined as n over n plus 1,"},{"Start":"00:08.280 ","End":"00:10.410","Text":"where n is any natural number."},{"Start":"00:10.410 ","End":"00:13.890","Text":"We have to show that the set A is bounded."},{"Start":"00:13.890 ","End":"00:16.350","Text":"Then we have to find 4 quantities,"},{"Start":"00:16.350 ","End":"00:17.670","Text":"the infimum, the supremum,"},{"Start":"00:17.670 ","End":"00:21.705","Text":"the maximum, and the minimum in so far as they exist."},{"Start":"00:21.705 ","End":"00:23.625","Text":"Start with part A."},{"Start":"00:23.625 ","End":"00:25.740","Text":"Start by writing out a few members of A,"},{"Start":"00:25.740 ","End":"00:27.825","Text":"just get a feeling of what\u0027s going on."},{"Start":"00:27.825 ","End":"00:30.855","Text":"I\u0027m taking the natural numbers to include 0."},{"Start":"00:30.855 ","End":"00:32.460","Text":"So 0 over 1,"},{"Start":"00:32.460 ","End":"00:34.170","Text":"1 over 2, 2 over 3,"},{"Start":"00:34.170 ","End":"00:38.280","Text":"and so on, 999 over 1000, etc."},{"Start":"00:38.280 ","End":"00:43.010","Text":"Just visually, it looks like 0 is the lower bound."},{"Start":"00:43.010 ","End":"00:45.710","Text":"I mean, we\u0027re never going to get any negative numbers this way"},{"Start":"00:45.710 ","End":"00:48.575","Text":"and looks like 1 is an upper bound."},{"Start":"00:48.575 ","End":"00:51.440","Text":"First of all, we show that 0 is a lower bound, well,"},{"Start":"00:51.440 ","End":"00:55.520","Text":"a_n is n over n plus 1 and it\u0027s bigger or equal to 0."},{"Start":"00:55.520 ","End":"00:57.365","Text":"It\u0027s just symbolic writing."},{"Start":"00:57.365 ","End":"01:00.870","Text":"n plus 1 is strictly bigger than 0 and some"},{"Start":"01:00.870 ","End":"01:05.300","Text":"think bigger or equal to 0 or bigger than 0 will be bigger or equal to 0."},{"Start":"01:05.300 ","End":"01:11.050","Text":"Also, n plus 1 is definitely positive."},{"Start":"01:11.050 ","End":"01:16.195","Text":"We can take this inequality and divide both sides by n plus 1."},{"Start":"01:16.195 ","End":"01:18.505","Text":"We get n over n plus 1,"},{"Start":"01:18.505 ","End":"01:23.210","Text":"which is a_n is less than n plus 1 over n plus 1, which is 1."},{"Start":"01:23.210 ","End":"01:25.820","Text":"From here we have that a_n is non-negative,"},{"Start":"01:25.820 ","End":"01:29.060","Text":"and from here we have that a_n is less than 1."},{"Start":"01:29.060 ","End":"01:32.240","Text":"That shows that we have a lower and upper bound."},{"Start":"01:32.240 ","End":"01:34.669","Text":"Then it\u0027s just called bounded."},{"Start":"01:34.669 ","End":"01:36.905","Text":"Now on to part B,"},{"Start":"01:36.905 ","End":"01:41.375","Text":"we have to find the infimum and the minimum first."},{"Start":"01:41.375 ","End":"01:44.375","Text":"We showed that 0 is a lower bound of A,"},{"Start":"01:44.375 ","End":"01:47.030","Text":"also 0 belongs to A."},{"Start":"01:47.030 ","End":"01:50.540","Text":"Now when you have a lower bound that belongs to the set,"},{"Start":"01:50.540 ","End":"01:52.685","Text":"then it\u0027s equal to the infimum,"},{"Start":"01:52.685 ","End":"01:54.845","Text":"the greatest lower bound."},{"Start":"01:54.845 ","End":"01:58.970","Text":"I mean, the greatest lower bound can\u0027t be bigger than 0"},{"Start":"01:58.970 ","End":"02:02.395","Text":"because then it wouldn\u0027t be a lower bound for 0."},{"Start":"02:02.395 ","End":"02:04.805","Text":"The infimum is equal to the minimum,"},{"Start":"02:04.805 ","End":"02:09.050","Text":"and that\u0027s what happens always when the infimum belongs to the set"},{"Start":"02:09.050 ","End":"02:13.770","Text":"and we still have the supremum and maximum to compute."},{"Start":"02:13.770 ","End":"02:17.235","Text":"Recall that a_n is n over n plus 1."},{"Start":"02:17.235 ","End":"02:19.340","Text":"Guess that 1 is the supremum."},{"Start":"02:19.340 ","End":"02:26.610","Text":"Remember we wrote out a few members and we had 1000 over 1001 is getting close to 1."},{"Start":"02:26.610 ","End":"02:27.855","Text":"That\u0027s a good guess."},{"Start":"02:27.855 ","End":"02:29.240","Text":"1 is an upper bound."},{"Start":"02:29.240 ","End":"02:30.545","Text":"We already showed that."},{"Start":"02:30.545 ","End":"02:33.995","Text":"But we also see from this inequality that"},{"Start":"02:33.995 ","End":"02:38.210","Text":"a_n can\u0027t be equal to 1 because it\u0027s strictly less than 1."},{"Start":"02:38.210 ","End":"02:40.505","Text":"So 1 is not in the set,"},{"Start":"02:40.505 ","End":"02:42.755","Text":"but we\u0027re still going to show that it\u0027s the supremum."},{"Start":"02:42.755 ","End":"02:44.950","Text":"It just won\u0027t be the maximum."},{"Start":"02:44.950 ","End":"02:50.354","Text":"We\u0027ll show that anything less than 1 call it s is not an upper bound,"},{"Start":"02:50.354 ","End":"02:52.865","Text":"ie, there is some a_n bigger than it."},{"Start":"02:52.865 ","End":"02:54.545","Text":"That\u0027s what we have to show."},{"Start":"02:54.545 ","End":"02:59.650","Text":"We want a_n to be bigger than s. That\u0027s the a_n is this."},{"Start":"02:59.650 ","End":"03:04.340","Text":"That\u0027s true if and only if just doing a bit of algebra here n plus 1 is positive,"},{"Start":"03:04.340 ","End":"03:05.885","Text":"so we can multiply out,"},{"Start":"03:05.885 ","End":"03:09.130","Text":"and then we can bring the ns over here."},{"Start":"03:09.130 ","End":"03:17.415","Text":"Then take n out the brackets and 1 minus s is positive because s is less than 1."},{"Start":"03:17.415 ","End":"03:27.035","Text":"We can divide by 1 minus s. This gives us a condition on n to make this thing true."},{"Start":"03:27.035 ","End":"03:29.210","Text":"Now, there is such an n,"},{"Start":"03:29.210 ","End":"03:33.260","Text":"but we need to explain why is natural number bigger than anything?"},{"Start":"03:33.260 ","End":"03:36.200","Text":"That is the Archimedean principle."},{"Start":"03:36.200 ","End":"03:41.140","Text":"Yeah, we\u0027re guaranteed an n bigger than any positive real number."},{"Start":"03:41.140 ","End":"03:45.580","Text":"Summarizing the last bit 1 is the supremum of A."},{"Start":"03:45.580 ","End":"03:50.675","Text":"But there\u0027s no maximum because the supremum is not in the set."},{"Start":"03:50.675 ","End":"03:54.150","Text":"That concludes this exercise."}],"ID":26606},{"Watched":false,"Name":"Exercise 7","Duration":"4m 11s","ChapterTopicVideoID":25803,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.260","Text":"In this exercise, we define a set a of real numbers as the set of all a_n,"},{"Start":"00:07.260 ","End":"00:10.950","Text":"where a_n is 1n^4 plus 2 and plus 1,"},{"Start":"00:10.950 ","End":"00:15.075","Text":"and n is a positive natural number."},{"Start":"00:15.075 ","End":"00:18.540","Text":"The plus here means that the 0 is excluded."},{"Start":"00:18.540 ","End":"00:25.124","Text":"We have to show first of all that a is bounded and then to find 4 things,"},{"Start":"00:25.124 ","End":"00:28.665","Text":"infimum, supremum, maximum and minimum of a,"},{"Start":"00:28.665 ","End":"00:30.720","Text":"insofar as they exist."},{"Start":"00:30.720 ","End":"00:36.485","Text":"Part a, we started out by writing a few members just to get an idea of what\u0027s going on."},{"Start":"00:36.485 ","End":"00:40.295","Text":"We have 1/4, 1/21, 1/88."},{"Start":"00:40.295 ","End":"00:42.200","Text":"I\u0027m not doing the computations,"},{"Start":"00:42.200 ","End":"00:45.680","Text":"but it looks like the numbers are getting smaller and smaller."},{"Start":"00:45.680 ","End":"00:47.840","Text":"The denominators are getting bigger and bigger and"},{"Start":"00:47.840 ","End":"00:50.120","Text":"it starts off with a 1/4 and going down,"},{"Start":"00:50.120 ","End":"00:56.210","Text":"so make an educated guess that 0 is a lower bound and a 1/4 is the upper bound."},{"Start":"00:56.210 ","End":"00:58.865","Text":"Let\u0027s show this more precisely,"},{"Start":"00:58.865 ","End":"01:01.895","Text":"a_n is 1^4 plus 2n plus 1."},{"Start":"01:01.895 ","End":"01:05.510","Text":"N is positive, so the denominator is positive."},{"Start":"01:05.510 ","End":"01:07.580","Text":"We have positive over positive,"},{"Start":"01:07.580 ","End":"01:12.635","Text":"which is positive and of course bigger than 0 implies bigger or equal to 0,"},{"Start":"01:12.635 ","End":"01:15.115","Text":"is what we need for the lower bound."},{"Start":"01:15.115 ","End":"01:21.020","Text":"We also have that until the fourth of 2n plus 1 is bigger or equal to 4,"},{"Start":"01:21.020 ","End":"01:22.430","Text":"because n is at least 1;"},{"Start":"01:22.430 ","End":"01:23.450","Text":"so this is at least 1,"},{"Start":"01:23.450 ","End":"01:27.235","Text":"this is at least 2, this is at least 1,1 plus 2 plus 1 is 4."},{"Start":"01:27.235 ","End":"01:29.680","Text":"If you take the reciprocal,"},{"Start":"01:29.680 ","End":"01:32.565","Text":"1this, is less than or equal to a 1/4."},{"Start":"01:32.565 ","End":"01:34.745","Text":"So, 1/4 is the upper bound."},{"Start":"01:34.745 ","End":"01:38.540","Text":"We have a lower bound of 0 and an upper bound of a 1/4"},{"Start":"01:38.540 ","End":"01:42.645","Text":"bounded above and below so it\u0027s bounded and that\u0027s part a."},{"Start":"01:42.645 ","End":"01:45.570","Text":"In part b, we have to find those quantities,"},{"Start":"01:45.570 ","End":"01:47.790","Text":"we\u0027ll start with the infimum and the minimum,"},{"Start":"01:47.790 ","End":"01:52.805","Text":"we\u0027ll make a guess just like here that the infimum is 0."},{"Start":"01:52.805 ","End":"01:55.060","Text":"These things go down to 0."},{"Start":"01:55.060 ","End":"01:57.815","Text":"So 0 is a lower bound."},{"Start":"01:57.815 ","End":"02:01.625","Text":"We just have to show that it\u0027s the greatest lower bound."},{"Start":"02:01.625 ","End":"02:06.890","Text":"0 is not in a because everything in a is strictly positive,"},{"Start":"02:06.890 ","End":"02:10.280","Text":"that\u0027s a pity because otherwise you could have finished here and said, yeah,"},{"Start":"02:10.280 ","End":"02:12.455","Text":"zeros the infimum and the minimum,"},{"Start":"02:12.455 ","End":"02:15.235","Text":"but no, so we have to work a bit more."},{"Start":"02:15.235 ","End":"02:19.100","Text":"Now, 0 is a lower bound and to show the list a greatest lower bound,"},{"Start":"02:19.100 ","End":"02:22.895","Text":"you have to show that anything greater than 0 is no longer a lower bound."},{"Start":"02:22.895 ","End":"02:24.500","Text":"We\u0027ll take s bigger than 0,"},{"Start":"02:24.500 ","End":"02:27.740","Text":"and we\u0027ll show that it\u0027s not a lower bound by showing that there is"},{"Start":"02:27.740 ","End":"02:31.040","Text":"some n such that a_n is less than s,"},{"Start":"02:31.040 ","End":"02:34.130","Text":"We want to find a solution to this inequality."},{"Start":"02:34.130 ","End":"02:36.255","Text":"This is a_n."},{"Start":"02:36.255 ","End":"02:43.830","Text":"This will be true if 1n^4 is less than s. Not saying that this implies this,"},{"Start":"02:43.830 ","End":"02:50.570","Text":"the other way around this implies this because n^4 is less than n^4 plus 2n plus 1."},{"Start":"02:50.570 ","End":"02:53.705","Text":"This fraction is bigger than this fraction."},{"Start":"02:53.705 ","End":"02:56.365","Text":"If this is less than s and this is less than s,"},{"Start":"02:56.365 ","End":"02:58.050","Text":"we\u0027re walking backwards here."},{"Start":"02:58.050 ","End":"03:02.840","Text":"Now if 1 is less than s then 1^4 will be less than s. Again,"},{"Start":"03:02.840 ","End":"03:05.080","Text":"because this is less than this."},{"Start":"03:05.080 ","End":"03:11.560","Text":"This will be true provided n is bigger than 1/s."},{"Start":"03:11.560 ","End":"03:14.150","Text":"This last part is actually if and only if."},{"Start":"03:14.150 ","End":"03:18.500","Text":"Anyway, this is true because of the Archimedean Principle,"},{"Start":"03:18.500 ","End":"03:22.720","Text":"there is an n bigger than any positive real number."},{"Start":"03:22.720 ","End":"03:25.535","Text":"0 is the infimum of a."},{"Start":"03:25.535 ","End":"03:28.550","Text":"But like we said, there\u0027s no minimum because 0 is not"},{"Start":"03:28.550 ","End":"03:31.685","Text":"a member of a that takes care of infimum of a minimum."},{"Start":"03:31.685 ","End":"03:34.610","Text":"Next, we have to find supremum and maximum."},{"Start":"03:34.610 ","End":"03:37.730","Text":"Recall that this is what a_n is."},{"Start":"03:37.730 ","End":"03:39.950","Text":"Now the educated guess like we saw before,"},{"Start":"03:39.950 ","End":"03:41.390","Text":"it looks like a 1/4,"},{"Start":"03:41.390 ","End":"03:44.705","Text":"which is the first member because it looks like it\u0027s decreasing."},{"Start":"03:44.705 ","End":"03:46.310","Text":"This is a supremum of a,"},{"Start":"03:46.310 ","End":"03:47.970","Text":"but we have to prove it."},{"Start":"03:47.970 ","End":"03:50.300","Text":"We showed that a 1/4 is an upper bound."},{"Start":"03:50.300 ","End":"03:54.770","Text":"The way to show that it\u0027s a supremum is just to note that it\u0027s in the set."},{"Start":"03:54.770 ","End":"03:57.410","Text":"Whenever an upper bound is in the set,"},{"Start":"03:57.410 ","End":"04:01.100","Text":"then it has to be the maximum and the supremum,"},{"Start":"04:01.100 ","End":"04:03.225","Text":"it\u0027s equal to a_1."},{"Start":"04:03.225 ","End":"04:09.335","Text":"The supremum of a and the maximum of a are both equal to 1/4."},{"Start":"04:09.335 ","End":"04:12.060","Text":"That concludes this exercise."}],"ID":26607},{"Watched":false,"Name":"Exercise 8","Duration":"5m 33s","ChapterTopicVideoID":25804,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.325","Text":"In this exercise, we have a set A of numbers and is defined like a sequence."},{"Start":"00:05.325 ","End":"00:09.090","Text":"For each natural number n, positive natural,"},{"Start":"00:09.090 ","End":"00:15.300","Text":"we define a_n to be n^4 plus n square plus 3 over 2n^4 plus 2n squared plus 8."},{"Start":"00:15.300 ","End":"00:18.870","Text":"Now, everything is okay here because the denominator is not 0."},{"Start":"00:18.870 ","End":"00:21.825","Text":"First of all, we have to show that a is bounded."},{"Start":"00:21.825 ","End":"00:24.720","Text":"Secondly, to find the infimum, supremum,"},{"Start":"00:24.720 ","End":"00:26.370","Text":"maximum, and minimum of a,"},{"Start":"00:26.370 ","End":"00:28.360","Text":"as far as they exist."},{"Start":"00:28.360 ","End":"00:34.820","Text":"We\u0027ll start by evaluating the first few members of the set A for n equals 1,"},{"Start":"00:34.820 ","End":"00:39.230","Text":"2, 3, and 4, especially if we write them as a decimal to 3 places."},{"Start":"00:39.230 ","End":"00:40.985","Text":"We can see the pattern here."},{"Start":"00:40.985 ","End":"00:47.380","Text":"It looks like this is increasing and it looks like it\u0027s heading towards 1/2."},{"Start":"00:47.380 ","End":"00:51.560","Text":"This is our guess at the first 1 is the least,"},{"Start":"00:51.560 ","End":"00:53.840","Text":"so it\u0027s a lower bound, 5/12,"},{"Start":"00:53.840 ","End":"00:57.110","Text":"and the upper bound is 1/2. Well, this is just a guess."},{"Start":"00:57.110 ","End":"00:59.600","Text":"We have to show this now. Let\u0027s see."},{"Start":"00:59.600 ","End":"01:01.895","Text":"We can show that this which is AN,"},{"Start":"01:01.895 ","End":"01:03.845","Text":"is really less than or equal to a half."},{"Start":"01:03.845 ","End":"01:10.320","Text":"Well, this is true if and only if 2 times this is equal to 1 times this."},{"Start":"01:10.320 ","End":"01:14.225","Text":"This is true if and only if it\u0027s just algebra."},{"Start":"01:14.225 ","End":"01:17.750","Text":"Now, subtract this from this side,"},{"Start":"01:17.750 ","End":"01:20.165","Text":"and this side is the same thing."},{"Start":"01:20.165 ","End":"01:22.970","Text":"We\u0027re left with 6 is less than or equal to 8,"},{"Start":"01:22.970 ","End":"01:25.260","Text":"which is always true and this is always true,"},{"Start":"01:25.260 ","End":"01:26.600","Text":"so this is always true,"},{"Start":"01:26.600 ","End":"01:29.595","Text":"so AN is less than or equal to 1/2,"},{"Start":"01:29.595 ","End":"01:31.200","Text":"and 1/2 is an upper bound."},{"Start":"01:31.200 ","End":"01:33.830","Text":"That will show that 5/12 is a lower bound."},{"Start":"01:33.830 ","End":"01:37.490","Text":"Well, this is bigger or equal to this if and only if, again,"},{"Start":"01:37.490 ","End":"01:42.380","Text":"just a bit of algebra and comes down to this."},{"Start":"01:42.380 ","End":"01:44.660","Text":"This is certainly true because n squared is bigger or"},{"Start":"01:44.660 ","End":"01:47.045","Text":"equal to 1 and this is bigger or equal to 2."},{"Start":"01:47.045 ","End":"01:49.490","Text":"The product is bigger or equal to 1 times 2."},{"Start":"01:49.490 ","End":"01:51.365","Text":"Because this is always true,"},{"Start":"01:51.365 ","End":"01:53.195","Text":"this is always true."},{"Start":"01:53.195 ","End":"01:57.905","Text":"Then bounded from above, bounded from below."},{"Start":"01:57.905 ","End":"02:01.825","Text":"Just simply bounded and that\u0027s part A."},{"Start":"02:01.825 ","End":"02:03.775","Text":"Now on to part B,"},{"Start":"02:03.775 ","End":"02:06.530","Text":"reminder what AN is."},{"Start":"02:06.530 ","End":"02:13.040","Text":"The infimum turns out to be easy because 5/12 not only is a lower bound of a,"},{"Start":"02:13.040 ","End":"02:15.140","Text":"but it belongs to the set a."},{"Start":"02:15.140 ","End":"02:18.140","Text":"Whenever a lower bound belongs to the set,"},{"Start":"02:18.140 ","End":"02:21.349","Text":"then it\u0027s equal to both the minimum and the infimum."},{"Start":"02:21.349 ","End":"02:25.250","Text":"We have minimum and infimum of 5/12."},{"Start":"02:25.250 ","End":"02:28.355","Text":"Now the supremum and the maximum."},{"Start":"02:28.355 ","End":"02:31.715","Text":"Was making a guess that 1/2 is a supremum of a."},{"Start":"02:31.715 ","End":"02:34.100","Text":"Let\u0027s see if we can prove this,"},{"Start":"02:34.100 ","End":"02:37.340","Text":"now we\u0027ve already shown that 1/2 is an upper bound."},{"Start":"02:37.340 ","End":"02:39.775","Text":"Thing is it may not be the least upper bound."},{"Start":"02:39.775 ","End":"02:44.330","Text":"Suppose the supremum is S and S is strictly less than 1/2."},{"Start":"02:44.330 ","End":"02:48.290","Text":"Going to reach a contradiction by showing that some,"},{"Start":"02:48.290 ","End":"02:51.560","Text":"AN 1 at least is bigger than S. This will be"},{"Start":"02:51.560 ","End":"02:53.420","Text":"a contradiction because if S is"},{"Start":"02:53.420 ","End":"02:56.915","Text":"an upper bound and you can\u0027t have any member bigger than it."},{"Start":"02:56.915 ","End":"03:00.530","Text":"This is AN and we want to find n such that AN is"},{"Start":"03:00.530 ","End":"03:04.295","Text":"bigger than S. What I\u0027m going to do is work backwards."},{"Start":"03:04.295 ","End":"03:08.090","Text":"I want this to be true and I want to simplify it."},{"Start":"03:08.090 ","End":"03:10.925","Text":"If I replace this and this by 0,"},{"Start":"03:10.925 ","End":"03:13.565","Text":"I\u0027m decreasing the numerator."},{"Start":"03:13.565 ","End":"03:19.880","Text":"If I replace 8 by 8n squared then I\u0027m increasing the denominator."},{"Start":"03:19.880 ","End":"03:23.645","Text":"Put the arrow here to show that this implies this."},{"Start":"03:23.645 ","End":"03:25.295","Text":"We\u0027re working our way backwards."},{"Start":"03:25.295 ","End":"03:27.620","Text":"If this is true, then this is true."},{"Start":"03:27.620 ","End":"03:30.980","Text":"I will find something eventually that is definitely true,"},{"Start":"03:30.980 ","End":"03:33.275","Text":"and then this will be true."},{"Start":"03:33.275 ","End":"03:38.850","Text":"Gathering this is n^4 over 2n^4 plus 10n squared."},{"Start":"03:38.850 ","End":"03:44.975","Text":"Then we can divide top and bottom by n squared and get this."},{"Start":"03:44.975 ","End":"03:49.895","Text":"Then we can multiply both sides by 2n squared plus 10,"},{"Start":"03:49.895 ","End":"03:51.295","Text":"which is positive,"},{"Start":"03:51.295 ","End":"03:54.935","Text":"we get this, then bring this over to the"},{"Start":"03:54.935 ","End":"04:00.110","Text":"left and take the n squared outside the brackets."},{"Start":"04:00.110 ","End":"04:02.570","Text":"What\u0027s in the bracket is 1 minus 2s."},{"Start":"04:02.570 ","End":"04:08.240","Text":"This is positive because S is less than 1/2."},{"Start":"04:08.240 ","End":"04:09.955","Text":"Let\u0027s see. Yeah, there it is."},{"Start":"04:09.955 ","End":"04:11.850","Text":"S is less than 1/2."},{"Start":"04:11.850 ","End":"04:13.845","Text":"1 minus 2s is positive,"},{"Start":"04:13.845 ","End":"04:17.780","Text":"so we can divide both sides by 1 minus 2s,"},{"Start":"04:17.780 ","End":"04:20.990","Text":"and get that n squared bigger than 10f of1 minus 2s,"},{"Start":"04:20.990 ","End":"04:23.855","Text":"and this derives from this."},{"Start":"04:23.855 ","End":"04:26.675","Text":"If we find n such that this is true,"},{"Start":"04:26.675 ","End":"04:28.490","Text":"then this will be true, this will be true, this will be true,"},{"Start":"04:28.490 ","End":"04:30.450","Text":"this will be true, this will be true, this will be true,"},{"Start":"04:30.450 ","End":"04:34.610","Text":"and then AN will be bigger than S. In principle,"},{"Start":"04:34.610 ","End":"04:39.620","Text":"there is n bigger than this because we can use the Archimedean Principle."},{"Start":"04:39.620 ","End":"04:43.700","Text":"This is a positive real number when a numerator is positive,"},{"Start":"04:43.700 ","End":"04:47.570","Text":"denominator is positive to the square root exists and it\u0027s positive."},{"Start":"04:47.570 ","End":"04:51.870","Text":"There is a natural number bigger than any positive real number."},{"Start":"04:51.870 ","End":"04:57.710","Text":"Since this exists, we found our N. This means that 1/2 is the supremum of a,"},{"Start":"04:57.710 ","End":"05:02.300","Text":"but there\u0027s no maximum since 1/2 is not a member of a."},{"Start":"05:02.300 ","End":"05:03.995","Text":"Show this right now."},{"Start":"05:03.995 ","End":"05:07.115","Text":"Suppose it is equal to 1/2."},{"Start":"05:07.115 ","End":"05:11.300","Text":"By cross-multiplying, we get that this is equal to this,"},{"Start":"05:11.300 ","End":"05:13.915","Text":"this gives us that 6 equals 8,"},{"Start":"05:13.915 ","End":"05:15.390","Text":"this is impossible,"},{"Start":"05:15.390 ","End":"05:17.010","Text":"so this is impossible,"},{"Start":"05:17.010 ","End":"05:19.140","Text":"so AN is never equal to 1/2."},{"Start":"05:19.140 ","End":"05:21.575","Text":"1/2 is a supremum,"},{"Start":"05:21.575 ","End":"05:24.425","Text":"but it doesn\u0027t belong to the set,"},{"Start":"05:24.425 ","End":"05:26.900","Text":"so it\u0027s not a maximum and there is no maximum."},{"Start":"05:26.900 ","End":"05:30.040","Text":"If there is a maximum, the maximum is equal to the supremum."},{"Start":"05:30.040 ","End":"05:33.910","Text":"That concludes this exercise."}],"ID":26608},{"Watched":false,"Name":"Exercise 9 part a","Duration":"3m 28s","ChapterTopicVideoID":25805,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"In this exercise, we have a set A,"},{"Start":"00:03.255 ","End":"00:05.670","Text":"which is the set of all a_n."},{"Start":"00:05.670 ","End":"00:08.910","Text":"N is a positive natural number,"},{"Start":"00:08.910 ","End":"00:11.610","Text":"and a_n is this expression."},{"Start":"00:11.610 ","End":"00:17.820","Text":"The square brackets is the integer part or the whole part of a number x."},{"Start":"00:17.820 ","End":"00:22.155","Text":"Basically, just throw out the fraction part after the decimal point."},{"Start":"00:22.155 ","End":"00:25.755","Text":"C is some positive real number,"},{"Start":"00:25.755 ","End":"00:29.510","Text":"and so a_n is the whole part of cn over n."},{"Start":"00:29.510 ","End":"00:33.845","Text":"The reason we took the positive natural numbers is you don\u0027t want to divide by 0."},{"Start":"00:33.845 ","End":"00:37.535","Text":"We have to show that A is bounded from above,"},{"Start":"00:37.535 ","End":"00:39.305","Text":"and to find its supremum,"},{"Start":"00:39.305 ","End":"00:41.705","Text":"and part B to show it\u0027s bounded from below,"},{"Start":"00:41.705 ","End":"00:43.550","Text":"and to find its infimum."},{"Start":"00:43.550 ","End":"00:46.275","Text":"Note that for non-negative x,"},{"Start":"00:46.275 ","End":"00:49.220","Text":"the integer part of x,"},{"Start":"00:49.220 ","End":"00:51.500","Text":"is always less than or equal to x."},{"Start":"00:51.500 ","End":"00:53.375","Text":"May not be true for negative numbers,"},{"Start":"00:53.375 ","End":"00:56.185","Text":"but certainly true for non-negative x."},{"Start":"00:56.185 ","End":"01:00.170","Text":"In particular, the whole part of cn over n,"},{"Start":"01:00.170 ","End":"01:03.305","Text":"is less than or equal to just plain cn over n,"},{"Start":"01:03.305 ","End":"01:08.060","Text":"which is c. We see that c is an upper bound of A."},{"Start":"01:08.060 ","End":"01:09.860","Text":"This is the expression a_n,"},{"Start":"01:09.860 ","End":"01:12.895","Text":"so all the elements of a are less than or equal to c."},{"Start":"01:12.895 ","End":"01:17.485","Text":"The claim is that this c is the least upper bound of A."},{"Start":"01:17.485 ","End":"01:22.565","Text":"What we have to show is that anything smaller than c is no longer an upper bound."},{"Start":"01:22.565 ","End":"01:24.410","Text":"If we take s less than c,"},{"Start":"01:24.410 ","End":"01:26.375","Text":"the way we show it\u0027s not an upper bound,"},{"Start":"01:26.375 ","End":"01:31.700","Text":"is to find at least 1 element of the set a that is 1 of the a_n is bigger"},{"Start":"01:31.700 ","End":"01:37.205","Text":"than s. Note that for any positive number really,"},{"Start":"01:37.205 ","End":"01:41.840","Text":"the integer part of x is always bigger than x minus 1."},{"Start":"01:41.840 ","End":"01:45.004","Text":"When we take off the fraction part after the decimal,"},{"Start":"01:45.004 ","End":"01:48.245","Text":"we\u0027re taking off less than 1. This is true."},{"Start":"01:48.245 ","End":"01:53.230","Text":"We\u0027re looking for a_n that\u0027s bigger than s. We have to find n. Now,"},{"Start":"01:53.230 ","End":"01:58.250","Text":"this will be true if the whole part of cn over n is bigger than s. Well,"},{"Start":"01:58.250 ","End":"02:01.130","Text":"this is if only if this is a_n."},{"Start":"02:01.130 ","End":"02:06.085","Text":"This is bigger than this by what we wrote here."},{"Start":"02:06.085 ","End":"02:07.730","Text":"The denominator is positive,"},{"Start":"02:07.730 ","End":"02:09.410","Text":"so the inequality remains."},{"Start":"02:09.410 ","End":"02:11.405","Text":"If this is bigger than s,"},{"Start":"02:11.405 ","End":"02:14.975","Text":"then this is bigger than s because I\u0027ve increased the numerator here."},{"Start":"02:14.975 ","End":"02:20.100","Text":"This will be true if c minus s is bigger than 1 over n,"},{"Start":"02:20.100 ","End":"02:21.540","Text":"or the 2 steps in 1."},{"Start":"02:21.540 ","End":"02:25.335","Text":"This is c minus 1 over n,"},{"Start":"02:25.335 ","End":"02:28.650","Text":"bring the 1 over n to the right bring the s to the left,"},{"Start":"02:28.650 ","End":"02:32.070","Text":"and we got c minus s bigger than 1 over n. A c minus s is"},{"Start":"02:32.070 ","End":"02:36.580","Text":"positive because s is less than c. This is true."},{"Start":"02:36.580 ","End":"02:38.785","Text":"It\u0027s actually if and only if."},{"Start":"02:38.785 ","End":"02:41.580","Text":"This is the same as switch,"},{"Start":"02:41.580 ","End":"02:42.900","Text":"these 2 around,"},{"Start":"02:42.900 ","End":"02:49.310","Text":"n is bigger than 1 over c minus s. We need n to be bigger than 1 over c"},{"Start":"02:49.310 ","End":"02:51.530","Text":"minus s. This is where we use"},{"Start":"02:51.530 ","End":"02:56.360","Text":"the Archimedean principle that says that for any positive real number,"},{"Start":"02:56.360 ","End":"02:59.480","Text":"we can find a natural number that\u0027s bigger than it."},{"Start":"02:59.480 ","End":"03:01.520","Text":"What we have is that for such an n,"},{"Start":"03:01.520 ","End":"03:02.875","Text":"we know it exists,"},{"Start":"03:02.875 ","End":"03:07.980","Text":"a_n is bigger than s because we get from here back to here."},{"Start":"03:08.270 ","End":"03:13.190","Text":"S is not an upper bound of A because at least 1 of the elements"},{"Start":"03:13.190 ","End":"03:17.980","Text":"in the set A is bigger than s. This shows that the supremum of A,"},{"Start":"03:17.980 ","End":"03:23.355","Text":"the least upper bound of A is c. That\u0027s the c we found earlier on."},{"Start":"03:23.355 ","End":"03:25.350","Text":"That\u0027s the c here."},{"Start":"03:25.350 ","End":"03:29.050","Text":"That\u0027s part A, and we\u0027ll take a break."}],"ID":26609},{"Watched":false,"Name":"Exercise 9 part b","Duration":"4m 15s","ChapterTopicVideoID":25806,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"We\u0027re back after the break,"},{"Start":"00:02.010 ","End":"00:03.170","Text":"we just did part a."},{"Start":"00:03.170 ","End":"00:04.500","Text":"Now part b."},{"Start":"00:04.500 ","End":"00:08.430","Text":"Remind that the set A is the set of all a_n as follows,"},{"Start":"00:08.430 ","End":"00:11.610","Text":"and we need to find the infimum of A."},{"Start":"00:11.610 ","End":"00:14.790","Text":"We also have to show that a is bounded from below first."},{"Start":"00:14.790 ","End":"00:17.595","Text":"Each of these is non-negative."},{"Start":"00:17.595 ","End":"00:19.710","Text":"There\u0027s no way this become negative."},{"Start":"00:19.710 ","End":"00:24.045","Text":"0 is a lower bound and the set is not empty,"},{"Start":"00:24.045 ","End":"00:26.220","Text":"so the infimum exists."},{"Start":"00:26.220 ","End":"00:30.675","Text":"We\u0027ll divide into 2 cases where c is a whole number and where c is not."},{"Start":"00:30.675 ","End":"00:33.300","Text":"This is Case 1 where c is a natural number,"},{"Start":"00:33.300 ","End":"00:36.220","Text":"actually, a positive natural number."},{"Start":"00:36.220 ","End":"00:40.370","Text":"In this case, c times n is also a natural number,"},{"Start":"00:40.370 ","End":"00:43.670","Text":"because product of 2 natural numbers is a natural number."},{"Start":"00:43.670 ","End":"00:47.330","Text":"We have that a_n is the whole part of cn,"},{"Start":"00:47.330 ","End":"00:49.010","Text":"but this is a whole number,"},{"Start":"00:49.010 ","End":"00:51.230","Text":"then the whole part is just itself."},{"Start":"00:51.230 ","End":"00:55.100","Text":"This is equal to c. This is true for all n. Really,"},{"Start":"00:55.100 ","End":"00:59.200","Text":"there\u0027s only 1 element in A and that is c, the singleton set."},{"Start":"00:59.200 ","End":"01:01.460","Text":"The infimum of the singleton set,"},{"Start":"01:01.460 ","End":"01:02.885","Text":"which is also the minimum,"},{"Start":"01:02.885 ","End":"01:04.805","Text":"is just that number c,"},{"Start":"01:04.805 ","End":"01:07.250","Text":"that was Case 1, that\u0027s the easy case."},{"Start":"01:07.250 ","End":"01:11.345","Text":"Now Case 2, c is not a natural number."},{"Start":"01:11.345 ","End":"01:15.650","Text":"Let k be the integer part of c, k, of course,"},{"Start":"01:15.650 ","End":"01:17.465","Text":"is a natural number,"},{"Start":"01:17.465 ","End":"01:21.095","Text":"and c is between k and k plus 1."},{"Start":"01:21.095 ","End":"01:22.790","Text":"Just think of an example."},{"Start":"01:22.790 ","End":"01:27.350","Text":"If c was 5.3, then it would be between 5 and 6,"},{"Start":"01:27.350 ","End":"01:31.330","Text":"strict inequality because c is not a whole number itself."},{"Start":"01:31.330 ","End":"01:34.710","Text":"If Epsilon be c minus k,"},{"Start":"01:34.710 ","End":"01:36.904","Text":"then Epsilon is a positive number."},{"Start":"01:36.904 ","End":"01:39.155","Text":"In fact, it\u0027s between 0 and 1."},{"Start":"01:39.155 ","End":"01:41.720","Text":"I also note that c is k plus Epsilon."},{"Start":"01:41.720 ","End":"01:42.890","Text":"Now, a_n,"},{"Start":"01:42.890 ","End":"01:51.935","Text":"which is integer part of cn over n is equal to cn is equal to k plus Epsilon times n,"},{"Start":"01:51.935 ","End":"01:55.980","Text":"because c is k plus Epsilon."},{"Start":"01:55.980 ","End":"01:59.195","Text":"I use a distributive law to expand the brackets."},{"Start":"01:59.195 ","End":"02:02.030","Text":"This is kn plus Epsilon n,"},{"Start":"02:02.030 ","End":"02:04.535","Text":"kn is a whole number."},{"Start":"02:04.535 ","End":"02:08.990","Text":"You can take the whole part out of the brackets."},{"Start":"02:08.990 ","End":"02:10.865","Text":"This is the rule to think about it."},{"Start":"02:10.865 ","End":"02:15.530","Text":"Let\u0027s say x was 5.3 and the whole part of x is 5,"},{"Start":"02:15.530 ","End":"02:17.060","Text":"if I add 4 to it,"},{"Start":"02:17.060 ","End":"02:19.480","Text":"we\u0027ll have the whole part of 9.3."},{"Start":"02:19.480 ","End":"02:21.410","Text":"The answer would be 9,"},{"Start":"02:21.410 ","End":"02:24.290","Text":"which is the same as affair that 4 to the 5."},{"Start":"02:24.290 ","End":"02:26.900","Text":"This kn comes out,"},{"Start":"02:26.900 ","End":"02:35.475","Text":"so it\u0027s kn plus the whole part of Epsilon n over n. This is equal to kn over n is k,"},{"Start":"02:35.475 ","End":"02:39.180","Text":"and here we have Epsilon n over n, the whole part here."},{"Start":"02:39.180 ","End":"02:40.790","Text":"This is a_n."},{"Start":"02:40.790 ","End":"02:44.165","Text":"Now, this part is non-negative,"},{"Start":"02:44.165 ","End":"02:51.390","Text":"so a_n is bigger or equal to k. This is true for all n. Note that when n equals 1,"},{"Start":"02:51.390 ","End":"02:57.015","Text":"we get equality because a_1 is k plus the whole part of Epsilon times 1 over 1."},{"Start":"02:57.015 ","End":"02:59.405","Text":"Now, Epsilon is between 0 and 1,"},{"Start":"02:59.405 ","End":"03:01.220","Text":"so the whole part is 0."},{"Start":"03:01.220 ","End":"03:03.245","Text":"This just comes out to be k,"},{"Start":"03:03.245 ","End":"03:06.890","Text":"so k is 1 of the members of a,"},{"Start":"03:06.890 ","End":"03:08.960","Text":"belongs to a and it\u0027s a lower bound,"},{"Start":"03:08.960 ","End":"03:11.825","Text":"and whenever we have a lower bound that\u0027s in the set,"},{"Start":"03:11.825 ","End":"03:14.590","Text":"then it\u0027s equal to the infimum and the minimum."},{"Start":"03:14.590 ","End":"03:16.200","Text":"The infimum of A,"},{"Start":"03:16.200 ","End":"03:18.480","Text":"which is the minimum of k, is k itself."},{"Start":"03:18.480 ","End":"03:25.425","Text":"I\u0027m going to remind you that k is the integer part of c. I want to put all this together."},{"Start":"03:25.425 ","End":"03:31.820","Text":"In Case 2, the infimum is integer part of c. In the first case,"},{"Start":"03:31.820 ","End":"03:36.800","Text":"in Case 1, we had that it was c itself, the infimum."},{"Start":"03:36.800 ","End":"03:40.400","Text":"We can combine these 2 cases. Let\u0027s see."},{"Start":"03:40.400 ","End":"03:45.015","Text":"Case 1, the infimum of A is c. Case 2,"},{"Start":"03:45.015 ","End":"03:48.910","Text":"infimum of A is the integer part of c. But, look,"},{"Start":"03:48.910 ","End":"03:56.300","Text":"in Case 1, c is equal to the integer part of c. Because if you have a whole number,"},{"Start":"03:56.300 ","End":"03:59.720","Text":"then the whole part is the same as the number itself."},{"Start":"03:59.720 ","End":"04:02.045","Text":"If c is a natural number,"},{"Start":"04:02.045 ","End":"04:05.615","Text":"then integer part of c is just c itself."},{"Start":"04:05.615 ","End":"04:09.260","Text":"In both cases, infimum of a is"},{"Start":"04:09.260 ","End":"04:16.360","Text":"the integer part of c. That concludes part b on this exercise."}],"ID":26610},{"Watched":false,"Name":"Exercise 10","Duration":"2m 21s","ChapterTopicVideoID":25807,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, the set A is given by the set of all a_n,"},{"Start":"00:05.370 ","End":"00:08.265","Text":"where a_n is n^5 minus n plus 4,"},{"Start":"00:08.265 ","End":"00:12.030","Text":"and then runs over all the natural numbers."},{"Start":"00:12.030 ","End":"00:14.490","Text":"We have to find the infimum, supremum, maximum,"},{"Start":"00:14.490 ","End":"00:17.190","Text":"and minimum of A as far as they exist."},{"Start":"00:17.190 ","End":"00:23.940","Text":"We\u0027ll start off by writing the first few members of A. I did it on the calculator,"},{"Start":"00:23.940 ","End":"00:25.440","Text":"you get 4, 34,"},{"Start":"00:25.440 ","End":"00:28.140","Text":"244 and then gets big."},{"Start":"00:28.140 ","End":"00:32.735","Text":"We\u0027ll say at the minimum of A is 4 and that a has no upper bound."},{"Start":"00:32.735 ","End":"00:34.295","Text":"But we have to prove these."},{"Start":"00:34.295 ","End":"00:35.840","Text":"It\u0027s just what it looks like."},{"Start":"00:35.840 ","End":"00:39.325","Text":"Start off with showing that 4 is a lower bound."},{"Start":"00:39.325 ","End":"00:45.345","Text":"The a_n is bigger or equal to 4 for all n. We need to prove this for all n,"},{"Start":"00:45.345 ","End":"00:48.860","Text":"and we can take n out the brackets here after we"},{"Start":"00:48.860 ","End":"00:53.300","Text":"subtract 4 from both sides and get n n^4 minus 1 bigger or equal to 0."},{"Start":"00:53.300 ","End":"00:55.460","Text":"This is true, then this is true."},{"Start":"00:55.460 ","End":"00:58.275","Text":"If n^4 is bigger or equal to 1,"},{"Start":"00:58.275 ","End":"01:02.480","Text":"this will be true since n is bigger or equal to 0,"},{"Start":"01:02.480 ","End":"01:05.690","Text":"and this will be true if n is bigger or equal to 1."},{"Start":"01:05.690 ","End":"01:07.460","Text":"But this is true, so this is true,"},{"Start":"01:07.460 ","End":"01:10.230","Text":"so this is true, so this is true."},{"Start":"01:10.340 ","End":"01:15.865","Text":"This shows us that the minimum and the infimum of A are all 4."},{"Start":"01:15.865 ","End":"01:18.220","Text":"When the lower bound is in the set,"},{"Start":"01:18.220 ","End":"01:19.945","Text":"that\u0027s the minimum and the infimum."},{"Start":"01:19.945 ","End":"01:23.490","Text":"Now, what about showing that it\u0027s not bounded from above."},{"Start":"01:23.490 ","End":"01:27.635","Text":"Showing that running large M is always an n,"},{"Start":"01:27.635 ","End":"01:31.100","Text":"such that a_n is bigger than M. However large you make M,"},{"Start":"01:31.100 ","End":"01:34.095","Text":"there\u0027s always an element of A bigger than it."},{"Start":"01:34.095 ","End":"01:37.920","Text":"Let\u0027s see if we can find a solution to this inequality,"},{"Start":"01:37.920 ","End":"01:43.875","Text":"n^5 minus n plus 4 is bigger than M. This will be true if this is true."},{"Start":"01:43.875 ","End":"01:46.325","Text":"This will be true if this is true,"},{"Start":"01:46.325 ","End":"01:49.340","Text":"because n is bigger or equal to 1."},{"Start":"01:49.340 ","End":"01:50.420","Text":"It\u0027s not if and only,"},{"Start":"01:50.420 ","End":"01:53.495","Text":"we just need to go in this direction, looking backwards."},{"Start":"01:53.495 ","End":"01:57.370","Text":"This will be true if n^4 is bigger than M plus 1."},{"Start":"01:57.370 ","End":"02:01.460","Text":"If we pick an integer bigger than the 4th root of n plus 1,"},{"Start":"02:01.460 ","End":"02:03.170","Text":"then this will be true."},{"Start":"02:03.170 ","End":"02:06.545","Text":"Such an n exists by the Archimedean Principle,"},{"Start":"02:06.545 ","End":"02:13.035","Text":"this is some positive real number and therefore there is a natural number bigger than it."},{"Start":"02:13.035 ","End":"02:18.995","Text":"So A has no upper bound and supremum and the maximum for A, don\u0027t exist."},{"Start":"02:18.995 ","End":"02:21.990","Text":"That concludes this exercise."}],"ID":26611},{"Watched":false,"Name":"Exercise 11","Duration":"4m 39s","ChapterTopicVideoID":25808,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.874","Text":"This exercise involves the Archimedean property of the real numbers."},{"Start":"00:04.874 ","End":"00:07.215","Text":"First part, to state and prove it."},{"Start":"00:07.215 ","End":"00:11.370","Text":"Part b is to prove that for any real Epsilon bigger than 0,"},{"Start":"00:11.370 ","End":"00:13.500","Text":"there exists a natural number n,"},{"Start":"00:13.500 ","End":"00:17.415","Text":"such that 1 over n is between 0 and Epsilon."},{"Start":"00:17.415 ","End":"00:21.480","Text":"Part c, to prove that if Alpha and Beta"},{"Start":"00:21.480 ","End":"00:23.370","Text":"are positive real numbers,"},{"Start":"00:23.370 ","End":"00:25.110","Text":"and Alpha\u0027s less than Beta,"},{"Start":"00:25.110 ","End":"00:31.455","Text":"then there exists a natural number n such that Alpha less than Alpha plus 1 over n,"},{"Start":"00:31.455 ","End":"00:32.715","Text":"less than Beta,"},{"Start":"00:32.715 ","End":"00:38.715","Text":"and also Alpha less than Beta minus 1 over n less than Beta."},{"Start":"00:38.715 ","End":"00:41.485","Text":"We start with part a."},{"Start":"00:41.485 ","End":"00:45.710","Text":"There are variants of the Archimedean property."},{"Start":"00:45.710 ","End":"00:48.020","Text":"Let\u0027s take the main one,"},{"Start":"00:48.020 ","End":"00:50.405","Text":"the one that I did in the tutorial,"},{"Start":"00:50.405 ","End":"00:54.230","Text":"which says that if a and b are positive real numbers,"},{"Start":"00:54.230 ","End":"00:58.880","Text":"then there is a natural number n such that n times a is bigger than b."},{"Start":"00:58.880 ","End":"01:01.370","Text":"Let\u0027s prove it by contradiction."},{"Start":"01:01.370 ","End":"01:05.240","Text":"We suppose on the contrary that na is less than or equal to"},{"Start":"01:05.240 ","End":"01:10.025","Text":"b for all n. Let K be the set of all na,"},{"Start":"01:10.025 ","End":"01:12.830","Text":"where n is a natural number."},{"Start":"01:12.830 ","End":"01:17.210","Text":"Then K is bounded above by b because"},{"Start":"01:17.210 ","End":"01:19.055","Text":"all the members of K,"},{"Start":"01:19.055 ","End":"01:21.090","Text":"the na are all less than or equal to b,"},{"Start":"01:21.090 ","End":"01:28.370","Text":"so b is an upper bound and K is not empty since a belongs to K. For example,"},{"Start":"01:28.370 ","End":"01:29.840","Text":"take n equals 1,"},{"Start":"01:29.840 ","End":"01:31.775","Text":"then na is 1a,"},{"Start":"01:31.775 ","End":"01:35.705","Text":"it belongs to K. By the completeness axiom,"},{"Start":"01:35.705 ","End":"01:37.925","Text":"K has a supremum,"},{"Start":"01:37.925 ","End":"01:39.470","Text":"let\u0027s call it s,"},{"Start":"01:39.470 ","End":"01:41.525","Text":"at least upper bound."},{"Start":"01:41.525 ","End":"01:45.500","Text":"That means that na is less than or equal to s"},{"Start":"01:45.500 ","End":"01:48.950","Text":"for all n in N because this is a typical element of"},{"Start":"01:48.950 ","End":"01:57.680","Text":"K. Now you can replace n by n plus 1 in this inequality and get that for all n in N,"},{"Start":"01:57.680 ","End":"02:05.185","Text":"n plus 1a is less than or equal to s. This gives na plus a less than or equal to s,"},{"Start":"02:05.185 ","End":"02:09.915","Text":"so na is less than or equal to s minus a."},{"Start":"02:09.915 ","End":"02:14.165","Text":"That means that s minus a is an upper bound of K,"},{"Start":"02:14.165 ","End":"02:16.700","Text":"but it\u0027s less than the supremum"},{"Start":"02:16.700 ","End":"02:21.190","Text":"and that\u0027s a contradiction which proves our claim."},{"Start":"02:21.190 ","End":"02:23.960","Text":"That was part a. Now part b,"},{"Start":"02:23.960 ","End":"02:25.325","Text":"I just copied it here."},{"Start":"02:25.325 ","End":"02:28.580","Text":"We have to prove that for any real positive Epsilon,"},{"Start":"02:28.580 ","End":"02:32.645","Text":"there was natural n such that 1 over n between 0 and Epsilon."},{"Start":"02:32.645 ","End":"02:36.800","Text":"Let a equal 1 and b equal 1 over Epsilon."},{"Start":"02:36.800 ","End":"02:39.615","Text":"Apply the Archimedean property,"},{"Start":"02:39.615 ","End":"02:45.650","Text":"na bigger than b for some n. What we get is that n times a,"},{"Start":"02:45.650 ","End":"02:47.390","Text":"which is 1, is bigger than b,"},{"Start":"02:47.390 ","End":"02:48.985","Text":"which is 1 over Epsilon,"},{"Start":"02:48.985 ","End":"02:51.220","Text":"n is bigger than 1 over Epsilon."},{"Start":"02:51.220 ","End":"02:54.920","Text":"Take the reciprocal and invert the direction of the inequality,"},{"Start":"02:54.920 ","End":"02:56.930","Text":"1 over n less than Epsilon."},{"Start":"02:56.930 ","End":"02:59.865","Text":"Of course, 1 over n is bigger than 0."},{"Start":"02:59.865 ","End":"03:01.200","Text":"That\u0027s b."},{"Start":"03:01.200 ","End":"03:05.765","Text":"Now, c, we have to prove that if Alpha is less than Beta,"},{"Start":"03:05.765 ","End":"03:08.335","Text":"and they\u0027re both real numbers,"},{"Start":"03:08.335 ","End":"03:11.075","Text":"Alpha and Beta are both positive,"},{"Start":"03:11.075 ","End":"03:12.320","Text":"then there exists n,"},{"Start":"03:12.320 ","End":"03:13.790","Text":"such that well, we have it written here."},{"Start":"03:13.790 ","End":"03:15.625","Text":"We already read it before."},{"Start":"03:15.625 ","End":"03:20.280","Text":"Define a to be equal to Beta minus Alpha."},{"Start":"03:20.280 ","End":"03:23.460","Text":"Don\u0027t confuse a with Alpha. They look similar."},{"Start":"03:23.460 ","End":"03:29.470","Text":"Let b equals 1. Beta minus Alpha is positive because Alpha is less than Beta."},{"Start":"03:29.470 ","End":"03:31.580","Text":"By the Archimedean property,"},{"Start":"03:31.580 ","End":"03:35.570","Text":"we can find n such that na is bigger than b."},{"Start":"03:35.570 ","End":"03:41.180","Text":"This means that n times Beta minus Alpha is bigger than 1."},{"Start":"03:41.180 ","End":"03:45.885","Text":"1 over n is less than Beta minus Alpha."},{"Start":"03:45.885 ","End":"03:49.515","Text":"Bring Alpha to this side, we get Alpha plus 1 over n is less than Beta."},{"Start":"03:49.515 ","End":"03:55.100","Text":"Similarly, the Alpha\u0027s less than Beta minus 1 over n. Get both of these."},{"Start":"03:55.100 ","End":"04:01.470","Text":"From the first one, we get that Alpha\u0027s less than Alpha plus 1 over n less than Beta."},{"Start":"04:01.470 ","End":"04:05.160","Text":"I mean, Alpha is obviously less than Alpha plus 1 over n. For the second one,"},{"Start":"04:05.160 ","End":"04:07.560","Text":"we can just put a Beta on the right-hand side and get Alpha"},{"Start":"04:07.560 ","End":"04:10.305","Text":"less than Beta minus 1 over n, less than Beta."},{"Start":"04:10.305 ","End":"04:13.365","Text":"That concludes part c. Remark,"},{"Start":"04:13.365 ","End":"04:17.655","Text":"b can be proved from c by letting Alpha equals 0,"},{"Start":"04:17.655 ","End":"04:19.530","Text":"and Beta equals Epsilon,"},{"Start":"04:19.530 ","End":"04:23.980","Text":"we get Alpha less than Alpha plus 1 over n is less than Beta."},{"Start":"04:23.980 ","End":"04:27.620","Text":"Then we get 0 less than 0 plus 1 over n less than Epsilon."},{"Start":"04:27.620 ","End":"04:33.465","Text":"This is just 1 over n. This is exactly what we have here."},{"Start":"04:33.465 ","End":"04:35.490","Text":"If you proved c first,"},{"Start":"04:35.490 ","End":"04:36.990","Text":"then you get b for free."},{"Start":"04:36.990 ","End":"04:39.370","Text":"That concludes this clip."}],"ID":26612},{"Watched":false,"Name":"Exercise 12","Duration":"4m 47s","ChapterTopicVideoID":25809,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.835","Text":"This exercise is in 2 parts."},{"Start":"00:02.835 ","End":"00:07.185","Text":"In part a, for any positive natural number n,"},{"Start":"00:07.185 ","End":"00:12.375","Text":"we define I_n to be the interval from n to infinity,"},{"Start":"00:12.375 ","End":"00:15.030","Text":"including the n. In other words,"},{"Start":"00:15.030 ","End":"00:18.060","Text":"this interval is a set of all x which are bigger or equal to"},{"Start":"00:18.060 ","End":"00:23.460","Text":"n. We have to prove that the intersection of all these intervals,"},{"Start":"00:23.460 ","End":"00:25.095","Text":"it\u0027s an infinite intersection,"},{"Start":"00:25.095 ","End":"00:27.405","Text":"that this is the empty set."},{"Start":"00:27.405 ","End":"00:28.920","Text":"If you want me to just spell it out,"},{"Start":"00:28.920 ","End":"00:30.720","Text":"the interval from 1 to infinity,"},{"Start":"00:30.720 ","End":"00:32.220","Text":"intersection 2 to infinity,"},{"Start":"00:32.220 ","End":"00:34.020","Text":"3 to infinity, etc,"},{"Start":"00:34.020 ","End":"00:35.295","Text":"is the empty set."},{"Start":"00:35.295 ","End":"00:36.870","Text":"In part b,"},{"Start":"00:36.870 ","End":"00:45.550","Text":"we define intervals J_n to be the half-closed interval from minus 1 over n to infinity."},{"Start":"00:45.550 ","End":"00:49.010","Text":"We have to prove that the interval from 0 to"},{"Start":"00:49.010 ","End":"00:51.350","Text":"infinity is contained"},{"Start":"00:51.350 ","End":"00:54.340","Text":"in the intersection of all the J_n."},{"Start":"00:54.340 ","End":"00:56.745","Text":"Well, let\u0027s start with part a,"},{"Start":"00:56.745 ","End":"00:59.270","Text":"we will show that this intersection is"},{"Start":"00:59.270 ","End":"01:01.640","Text":"the empty set is by showing that"},{"Start":"01:01.640 ","End":"01:03.440","Text":"for any real number y,"},{"Start":"01:03.440 ","End":"01:06.715","Text":"y does not belong to this intersection."},{"Start":"01:06.715 ","End":"01:09.240","Text":"There is nothing in this intersection,"},{"Start":"01:09.240 ","End":"01:10.650","Text":"so it\u0027s the empty set."},{"Start":"01:10.650 ","End":"01:12.705","Text":"We\u0027ll divide into 2 cases;"},{"Start":"01:12.705 ","End":"01:16.594","Text":"the first case, y is less than or equal to 0,"},{"Start":"01:16.594 ","End":"01:21.880","Text":"then y does not belong to the interval from 1 to infinity,"},{"Start":"01:21.880 ","End":"01:23.580","Text":"which is I_1,"},{"Start":"01:23.580 ","End":"01:28.070","Text":"so it doesn\u0027t belong to the intersection of all the I_n."},{"Start":"01:28.070 ","End":"01:29.540","Text":"If it doesn\u0027t belong to one of them,"},{"Start":"01:29.540 ","End":"01:32.480","Text":"that\u0027s enough that it doesn\u0027t belong to the intersection."},{"Start":"01:32.480 ","End":"01:35.945","Text":"That\u0027s the case where y is less than or equal to 0."},{"Start":"01:35.945 ","End":"01:38.240","Text":"Now the case y bigger than 0,"},{"Start":"01:38.240 ","End":"01:41.725","Text":"again we have to show that y does not belong to the intersection."},{"Start":"01:41.725 ","End":"01:43.895","Text":"We\u0027ll use the Archimedean property."},{"Start":"01:43.895 ","End":"01:48.230","Text":"There is some natural number n naught bigger than y and this"},{"Start":"01:48.230 ","End":"01:50.510","Text":"means that y is not in the interval"},{"Start":"01:50.510 ","End":"01:52.850","Text":"from n naught to infinity,"},{"Start":"01:52.850 ","End":"01:56.530","Text":"which is what we call I sub n naught."},{"Start":"01:56.630 ","End":"02:00.770","Text":"Again, if y doesn\u0027t belong to one of the i\u0027s,"},{"Start":"02:00.770 ","End":"02:03.520","Text":"doesn\u0027t belong to the intersection of all the i\u0027s."},{"Start":"02:03.520 ","End":"02:05.675","Text":"That concludes part a,"},{"Start":"02:05.675 ","End":"02:07.970","Text":"that there is nothing in the intersection"},{"Start":"02:07.970 ","End":"02:09.080","Text":"of all the I_n."},{"Start":"02:09.080 ","End":"02:10.745","Text":"In other words, it\u0027s the empty set."},{"Start":"02:10.745 ","End":"02:13.945","Text":"Now we\u0027ll go on to part b."},{"Start":"02:13.945 ","End":"02:20.035","Text":"Here we have to show that this interval is in the intersection of the J_n."},{"Start":"02:20.035 ","End":"02:24.020","Text":"Just to remind you, this is what J_n is."},{"Start":"02:24.080 ","End":"02:28.385","Text":"If y is in 0 infinity,"},{"Start":"02:28.385 ","End":"02:31.160","Text":"which means that y is bigger or equal to 0,"},{"Start":"02:31.160 ","End":"02:39.400","Text":"then y is bigger or equal to minus 1 over n. This is true for all natural n. Now,"},{"Start":"02:39.400 ","End":"02:43.550","Text":"this exactly means that y belongs to J_n."},{"Start":"02:43.550 ","End":"02:47.270","Text":"That\u0027s what it is to be bigger or equal to minus 1 over n. This is true"},{"Start":"02:47.270 ","End":"02:51.120","Text":"for all n. If it\u0027s true and it\u0027s in J_n for all n,"},{"Start":"02:51.120 ","End":"02:55.910","Text":"then it belongs to the intersection overall n from 1 to infinity."},{"Start":"02:55.910 ","End":"02:57.290","Text":"That concludes part b."},{"Start":"02:57.290 ","End":"03:00.680","Text":"But don\u0027t go, I\u0027m going to give you a bonus part."},{"Start":"03:00.680 ","End":"03:04.235","Text":"We proved that this is a subset of this."},{"Start":"03:04.235 ","End":"03:06.530","Text":"We can actually prove that they\u0027re equal."},{"Start":"03:06.530 ","End":"03:12.230","Text":"We can prove reverse containment that this is a subset of this that will prove equality."},{"Start":"03:12.230 ","End":"03:13.910","Text":"That\u0027s what we\u0027re going to do now."},{"Start":"03:13.910 ","End":"03:18.380","Text":"We have to show that if y is in the intersection of the J_n,"},{"Start":"03:18.380 ","End":"03:22.010","Text":"then it\u0027s in the interval 0 to infinity."},{"Start":"03:22.010 ","End":"03:23.540","Text":"Do it by contradiction."},{"Start":"03:23.540 ","End":"03:29.200","Text":"Suppose y does not belong to 0 infinity,"},{"Start":"03:29.200 ","End":"03:34.775","Text":"that means that y is negative and this means that minus y is positive."},{"Start":"03:34.775 ","End":"03:36.980","Text":"By the previous exercise,"},{"Start":"03:36.980 ","End":"03:38.780","Text":"one of them that we used a lot,"},{"Start":"03:38.780 ","End":"03:45.905","Text":"there is some natural number n such that 1 over n is less than this positive minus y."},{"Start":"03:45.905 ","End":"03:47.690","Text":"Let\u0027s see what we get."},{"Start":"03:47.690 ","End":"03:53.000","Text":"We get that 1 over n is less than minus y,"},{"Start":"03:53.000 ","End":"03:57.280","Text":"and this means that y is less than minus 1 over n."},{"Start":"03:57.280 ","End":"04:01.790","Text":"Because we can multiply both sides by minus 1,"},{"Start":"04:01.790 ","End":"04:05.785","Text":"that reverses the inequality and then reverse sides so we get this."},{"Start":"04:05.785 ","End":"04:10.840","Text":"This means that y does not belong to J_n because J_n"},{"Start":"04:10.840 ","End":"04:15.975","Text":"is where y is bigger or equal to minus 1 over n. Whoops,"},{"Start":"04:15.975 ","End":"04:17.295","Text":"I wrote this twice."},{"Start":"04:17.295 ","End":"04:22.480","Text":"Y is not in J_n. If y is not in J_n for this particular n,"},{"Start":"04:22.480 ","End":"04:24.925","Text":"then it\u0027s not in the intersection,"},{"Start":"04:24.925 ","End":"04:28.480","Text":"just has to miss one of them and it\u0027s not in the intersection."},{"Start":"04:28.480 ","End":"04:30.830","Text":"To be in the intersection has to be in all of them,"},{"Start":"04:30.830 ","End":"04:32.770","Text":"and that\u0027s a contradiction because we"},{"Start":"04:32.770 ","End":"04:35.485","Text":"started off from assuming that y is in the intersection."},{"Start":"04:35.485 ","End":"04:39.820","Text":"This contradiction came from y naught in this interval,"},{"Start":"04:39.820 ","End":"04:42.395","Text":"so y is in this interval."},{"Start":"04:42.395 ","End":"04:44.660","Text":"That concludes this exercise,"},{"Start":"04:44.660 ","End":"04:47.460","Text":"including a bonus part."}],"ID":26613},{"Watched":false,"Name":"Exercise 13","Duration":"2m 27s","ChapterTopicVideoID":25810,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"In this exercise, for each natural number n,"},{"Start":"00:03.900 ","End":"00:10.335","Text":"we define an interval I_n as the open interval from minus 1 to 1."},{"Start":"00:10.335 ","End":"00:13.440","Text":"We have to prove that the intersection of"},{"Start":"00:13.440 ","End":"00:18.585","Text":"all these intervals is the singleton set containing just 0."},{"Start":"00:18.585 ","End":"00:22.575","Text":"Note, that each of these intervals I_n contains 0,"},{"Start":"00:22.575 ","End":"00:26.325","Text":"so 0 is in the intersection,"},{"Start":"00:26.325 ","End":"00:31.745","Text":"so the singleton set 0 is a subset of the intersection."},{"Start":"00:31.745 ","End":"00:34.910","Text":"To show equality, we need to prove"},{"Start":"00:34.910 ","End":"00:41.600","Text":"the reverse containment that the intersection is a subset of the singleton set 0."},{"Start":"00:41.600 ","End":"00:44.375","Text":"Suppose x is in this intersection,"},{"Start":"00:44.375 ","End":"00:46.460","Text":"we have to show that x is in here,"},{"Start":"00:46.460 ","End":"00:48.440","Text":"in other words that x is 0."},{"Start":"00:48.440 ","End":"00:50.030","Text":"We\u0027ll do this by contradiction,"},{"Start":"00:50.030 ","End":"00:52.960","Text":"and we suppose on the contrary that x is not 0."},{"Start":"00:52.960 ","End":"00:55.035","Text":"Then we have 2 possibilities,"},{"Start":"00:55.035 ","End":"00:59.765","Text":"either x is bigger than 0 or x is less than 0."},{"Start":"00:59.765 ","End":"01:01.340","Text":"If x is bigger than 0,"},{"Start":"01:01.340 ","End":"01:03.530","Text":"then also 1/ x is bigger than 0,"},{"Start":"01:03.530 ","End":"01:08.330","Text":"and we can apply the Archimedean property to find a natural number m,"},{"Start":"01:08.330 ","End":"01:11.075","Text":"which is bigger than 1/x."},{"Start":"01:11.075 ","End":"01:14.315","Text":"Just reversing this inequality,"},{"Start":"01:14.315 ","End":"01:17.625","Text":"we can get that x is bigger than 1/m,"},{"Start":"01:17.625 ","End":"01:23.540","Text":"so x does not belong to the interval from minus 1/m to 1/m."},{"Start":"01:23.540 ","End":"01:27.620","Text":"So x does not belong to the interval I_m,"},{"Start":"01:27.620 ","End":"01:30.635","Text":"and so, it does not belong to the intersection."},{"Start":"01:30.635 ","End":"01:32.450","Text":"If it doesn\u0027t belong to even one of them,"},{"Start":"01:32.450 ","End":"01:34.070","Text":"it doesn\u0027t belong to the intersection,"},{"Start":"01:34.070 ","End":"01:40.015","Text":"that\u0027s a contradiction because we assumed that x is in the intersection."},{"Start":"01:40.015 ","End":"01:43.070","Text":"Now, second case where x is negative,"},{"Start":"01:43.070 ","End":"01:46.670","Text":"then we get that minus x is positive,"},{"Start":"01:46.670 ","End":"01:49.310","Text":"and then minus 1/x is positive."},{"Start":"01:49.310 ","End":"01:51.470","Text":"So by the Archimedean property,"},{"Start":"01:51.470 ","End":"01:56.840","Text":"we can find natural m such that m is bigger than minus 1/x."},{"Start":"01:56.840 ","End":"02:01.325","Text":"If we multiply by x which is negative,"},{"Start":"02:01.325 ","End":"02:06.140","Text":"we get the xm is less than minus 1, and from this,"},{"Start":"02:06.140 ","End":"02:11.750","Text":"we can divide by positive m to get that x is less than minus 1/m."},{"Start":"02:11.750 ","End":"02:15.625","Text":"So once again, x is not in the interval I_m,"},{"Start":"02:15.625 ","End":"02:17.970","Text":"and therefore it\u0027s not in the intersection,"},{"Start":"02:17.970 ","End":"02:20.160","Text":"and again, we get a contradiction."},{"Start":"02:20.160 ","End":"02:24.050","Text":"This contradiction shows that x is equal to 0,"},{"Start":"02:24.050 ","End":"02:25.370","Text":"and that\u0027s what we needed to show,"},{"Start":"02:25.370 ","End":"02:27.690","Text":"and we are done."}],"ID":26614},{"Watched":false,"Name":"The Well-Ordering Principle","Duration":"2m 33s","ChapterTopicVideoID":25814,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"Now yet another principle,"},{"Start":"00:03.330 ","End":"00:05.835","Text":"something called the well-ordering principle."},{"Start":"00:05.835 ","End":"00:09.190","Text":"This applies to the natural numbers. It\u0027s a theorem."},{"Start":"00:09.190 ","End":"00:14.215","Text":"Every non-empty subset of the natural numbers has a least element."},{"Start":"00:14.215 ","End":"00:17.020","Text":"Note by the way, that if they\u0027re natural numbers,"},{"Start":"00:17.020 ","End":"00:20.980","Text":"I said integers, and it wouldn\u0027t be true because there is no least integer."},{"Start":"00:20.980 ","End":"00:24.535","Text":"It\u0027s not immediately obvious that this is true."},{"Start":"00:24.535 ","End":"00:27.285","Text":"In fact, we should prove it."},{"Start":"00:27.285 ","End":"00:30.610","Text":"This is equivalent to saying that every set of"},{"Start":"00:30.610 ","End":"00:34.375","Text":"natural numbers that hasn\u0027t got a least element is empty."},{"Start":"00:34.375 ","End":"00:37.235","Text":"We prove this it\u0027s equivalent to this."},{"Start":"00:37.235 ","End":"00:39.970","Text":"Let A be such a set,"},{"Start":"00:39.970 ","End":"00:42.600","Text":"I mean that it hasn\u0027t got at least element."},{"Start":"00:42.600 ","End":"00:45.290","Text":"We\u0027ll prove it\u0027s empty using induction."},{"Start":"00:45.290 ","End":"00:48.260","Text":"We\u0027ll define a property P of n,"},{"Start":"00:48.260 ","End":"00:51.635","Text":"which says that k is not a member of A,"},{"Start":"00:51.635 ","End":"00:53.060","Text":"for k equals 0, 1,"},{"Start":"00:53.060 ","End":"00:55.010","Text":"2, and so on up to n,"},{"Start":"00:55.010 ","End":"00:58.220","Text":"because I\u0027m assuming that the natural numbers include 0, if it doesn\u0027t,"},{"Start":"00:58.220 ","End":"01:03.625","Text":"then we\u0027ll take it from 1 to n. What we need to prove,"},{"Start":"01:03.625 ","End":"01:08.765","Text":"according to principle of induction is that P of 0 is true,"},{"Start":"01:08.765 ","End":"01:10.925","Text":"and if P of n is true,"},{"Start":"01:10.925 ","End":"01:13.405","Text":"then P of n plus 1 is true."},{"Start":"01:13.405 ","End":"01:17.445","Text":"P of 0 just means that 0 doesn\u0027t belong to A,"},{"Start":"01:17.445 ","End":"01:19.980","Text":"because from 0 up to 0 is just 0."},{"Start":"01:19.980 ","End":"01:23.010","Text":"This is true, because if 0 was an A,"},{"Start":"01:23.010 ","End":"01:24.900","Text":"0 would be the least element,"},{"Start":"01:24.900 ","End":"01:27.950","Text":"and we say that it hasn\u0027t got a least element."},{"Start":"01:27.950 ","End":"01:30.965","Text":"The second part, if P of n is true,"},{"Start":"01:30.965 ","End":"01:32.990","Text":"then 0 does not belong to A,"},{"Start":"01:32.990 ","End":"01:34.490","Text":"1 does not belong to A,"},{"Start":"01:34.490 ","End":"01:37.430","Text":"and so on up to n does not belong to A."},{"Start":"01:37.430 ","End":"01:41.240","Text":"Well, it follows that n plus 1 is also not in A, because if it was,"},{"Start":"01:41.240 ","End":"01:42.740","Text":"it would be the least element,"},{"Start":"01:42.740 ","End":"01:46.340","Text":"because all the ones below n plus 1 are not in A."},{"Start":"01:46.340 ","End":"01:49.085","Text":"0 doesn\u0027t belong to A, and so on."},{"Start":"01:49.085 ","End":"01:52.160","Text":"N doesn\u0027t belong to A, and n plus 1 does not belong to A,"},{"Start":"01:52.160 ","End":"01:56.020","Text":"and that is exactly P of n plus 1."},{"Start":"01:56.020 ","End":"01:59.540","Text":"By induction, P of n is true for all n,"},{"Start":"01:59.540 ","End":"02:03.335","Text":"which means that n does not belong to A. I mean,"},{"Start":"02:03.335 ","End":"02:04.760","Text":"zero doesn\u0027t belong to A and so on."},{"Start":"02:04.760 ","End":"02:07.250","Text":"In particular n does not belong to A."},{"Start":"02:07.250 ","End":"02:12.290","Text":"This is true for all n. If n doesn\u0027t belong to A for all n,"},{"Start":"02:12.290 ","End":"02:14.180","Text":"it means that A is empty,"},{"Start":"02:14.180 ","End":"02:15.905","Text":"and that concludes the proof,"},{"Start":"02:15.905 ","End":"02:19.175","Text":"a remark that it works the other way around also."},{"Start":"02:19.175 ","End":"02:23.185","Text":"Just as we proved the well-ordering principle from the principle of induction,"},{"Start":"02:23.185 ","End":"02:24.770","Text":"we can do it the other way around."},{"Start":"02:24.770 ","End":"02:28.040","Text":"The principle of induction can be proved from the well-ordering principles."},{"Start":"02:28.040 ","End":"02:31.100","Text":"The 2 principles are in fact equivalent."},{"Start":"02:31.100 ","End":"02:33.900","Text":"That concludes this clip."}],"ID":26618},{"Watched":false,"Name":"Exercise 14","Duration":"3m 43s","ChapterTopicVideoID":25811,"CourseChapterTopicPlaylistID":246308,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we have to prove that for every real number c,"},{"Start":"00:05.220 ","End":"00:09.660","Text":"there exists a unique integer m such"},{"Start":"00:09.660 ","End":"00:14.910","Text":"that c is sandwiched between m and m plus 1, like so."},{"Start":"00:14.910 ","End":"00:19.905","Text":"By the way, this unique m is defined as the floor function of x,"},{"Start":"00:19.905 ","End":"00:21.795","Text":"also denoted this way."},{"Start":"00:21.795 ","End":"00:24.975","Text":"Here\u0027s something that I copied from the Wikipedia."},{"Start":"00:24.975 ","End":"00:29.010","Text":"Okay, we\u0027ll prove the existence first and then we\u0027ll get to the uniqueness."},{"Start":"00:29.010 ","End":"00:31.590","Text":"I will split the existence up into 2 cases."},{"Start":"00:31.590 ","End":"00:33.900","Text":"Case 1, where c is bigger or equal to 0"},{"Start":"00:33.900 ","End":"00:35.440","Text":"and then c less than 0."},{"Start":"00:35.440 ","End":"00:38.970","Text":"We use the Archimedean property to conclude that there"},{"Start":"00:38.970 ","End":"00:43.765","Text":"is some natural n. A natural number is also an integer,"},{"Start":"00:43.765 ","End":"00:47.150","Text":"such that n is bigger than c. To be precise,"},{"Start":"00:47.150 ","End":"00:50.750","Text":"the Archimedean property relates to when c is bigger than 0."},{"Start":"00:50.750 ","End":"00:52.460","Text":"But if c is equal to 0,"},{"Start":"00:52.460 ","End":"00:55.505","Text":"then just choose n equals 1 and that will do."},{"Start":"00:55.505 ","End":"00:58.500","Text":"Now choose the least such n,"},{"Start":"00:58.500 ","End":"01:00.765","Text":"we have the Well-Ordering principle,"},{"Start":"01:00.765 ","End":"01:04.550","Text":"then a non-empty set of natural numbers"},{"Start":"01:04.550 ","End":"01:06.785","Text":"has a smallest member."},{"Start":"01:06.785 ","End":"01:09.830","Text":"We\u0027ll take the least n that satisfies"},{"Start":"01:09.830 ","End":"01:11.585","Text":"the property that n is bigger than"},{"Start":"01:11.585 ","End":"01:16.790","Text":"c. Because it\u0027s the least n minus 1 is no longer bigger than c,"},{"Start":"01:16.790 ","End":"01:18.440","Text":"it\u0027s less than or equal to c."},{"Start":"01:18.440 ","End":"01:19.820","Text":"We have c so much"},{"Start":"01:19.820 ","End":"01:23.590","Text":"between n minus 1 and n. That\u0027s almost what we want."},{"Start":"01:23.590 ","End":"01:26.150","Text":"What we have to do is just relabel a bit,"},{"Start":"01:26.150 ","End":"01:28.415","Text":"take m equal n minus 1,"},{"Start":"01:28.415 ","End":"01:34.460","Text":"and then we have m is less than or equal to c is less than m plus 1 as required."},{"Start":"01:34.460 ","End":"01:38.285","Text":"Now case 2, where c is negative,"},{"Start":"01:38.285 ","End":"01:41.155","Text":"then minus c is positive."},{"Start":"01:41.155 ","End":"01:46.085","Text":"We can find some natural number n bigger than minus c,"},{"Start":"01:46.085 ","End":"01:48.380","Text":"again by the Archimedean property,"},{"Start":"01:48.380 ","End":"01:50.285","Text":"and then if I add n,"},{"Start":"01:50.285 ","End":"01:53.410","Text":"I got c plus n is bigger than 0."},{"Start":"01:53.410 ","End":"02:00.790","Text":"Then we can use case 1 to find an integer m such that c plus n is between m and m plus 1."},{"Start":"02:00.790 ","End":"02:04.850","Text":"Now subtract n from all 3 sides and we have m"},{"Start":"02:04.850 ","End":"02:09.325","Text":"minus n less than or equal to c less than or equal to m minus n plus 1."},{"Start":"02:09.325 ","End":"02:12.695","Text":"If we call this m prime,"},{"Start":"02:12.695 ","End":"02:17.270","Text":"then we have m prime less than or equal to c,"},{"Start":"02:17.270 ","End":"02:19.055","Text":"less than m prime plus 1."},{"Start":"02:19.055 ","End":"02:22.120","Text":"This m prime is the m that we\u0027re looking for."},{"Start":"02:22.120 ","End":"02:25.340","Text":"That\u0027s the existence in both cases."},{"Start":"02:25.340 ","End":"02:29.800","Text":"Now, the proof of the uniqueness for all cases combined."},{"Start":"02:29.800 ","End":"02:32.505","Text":"Given c in r,"},{"Start":"02:32.505 ","End":"02:34.500","Text":"suppose we have 2 different m\u0027s,"},{"Start":"02:34.500 ","End":"02:35.835","Text":"m1 and m2,"},{"Start":"02:35.835 ","End":"02:37.845","Text":"both of which satisfy the condition,"},{"Start":"02:37.845 ","End":"02:43.105","Text":"i.e m1 less than or equal to c less than m1 plus 1 and similarly for m2."},{"Start":"02:43.105 ","End":"02:47.990","Text":"What we can do is we can say that this is less than or equal to this,"},{"Start":"02:47.990 ","End":"02:52.385","Text":"which is less than this and we get this condition."},{"Start":"02:52.385 ","End":"02:55.580","Text":"Similarly, this less than or equal to c,"},{"Start":"02:55.580 ","End":"02:57.095","Text":"less than m1 plus 1,"},{"Start":"02:57.095 ","End":"02:59.620","Text":"so we have this inequality."},{"Start":"02:59.620 ","End":"03:02.190","Text":"Eliminate the c from each of these,"},{"Start":"03:02.190 ","End":"03:08.095","Text":"so we have m1 less than m2 plus 1 and m2 less than m1 plus 1."},{"Start":"03:08.095 ","End":"03:09.140","Text":"From each of these,"},{"Start":"03:09.140 ","End":"03:12.970","Text":"we can get an inequality in m1 minus m2."},{"Start":"03:12.970 ","End":"03:16.410","Text":"On the 1 hand, m1 minus m2 is less than 1 from"},{"Start":"03:16.410 ","End":"03:20.010","Text":"here and from here bring the m2 over and the 1 over."},{"Start":"03:20.010 ","End":"03:23.520","Text":"But m1 minus m2 bigger than minus 1,"},{"Start":"03:23.520 ","End":"03:26.855","Text":"so m1 minus m2 is an integer,"},{"Start":"03:26.855 ","End":"03:30.020","Text":"and because it\u0027s less than 1 and bigger than minus 1,"},{"Start":"03:30.020 ","End":"03:32.089","Text":"it can only possibly be 0."},{"Start":"03:32.089 ","End":"03:34.550","Text":"That\u0027s the only integer in that range."},{"Start":"03:34.550 ","End":"03:36.625","Text":"If this minus this is 0,"},{"Start":"03:36.625 ","End":"03:44.500","Text":"then m1 equals m2 and that proves the uniqueness and that concludes this exercise."}],"ID":26615}],"Thumbnail":null,"ID":246308},{"Name":"Mathematical Induction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is Induction","Duration":"10m 19s","ChapterTopicVideoID":25834,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"A new topic, Mathematical Induction,"},{"Start":"00:03.105 ","End":"00:06.900","Text":"also known as proof by induction. What is it?"},{"Start":"00:06.900 ","End":"00:12.000","Text":"It\u0027s a method of proof to prove statements about the natural numbers."},{"Start":"00:12.000 ","End":"00:15.630","Text":"The question whether 0 is or isn\u0027t a natural number,"},{"Start":"00:15.630 ","End":"00:18.480","Text":"let\u0027s say that it isn\u0027t and if we need it to be,"},{"Start":"00:18.480 ","End":"00:20.970","Text":"then we\u0027ll state otherwise."},{"Start":"00:20.970 ","End":"00:28.430","Text":"Now usually have a claim of the form every natural number n satisfies property"},{"Start":"00:28.430 ","End":"00:31.545","Text":"so and so which is some expression involving"},{"Start":"00:31.545 ","End":"00:35.860","Text":"n. P of n is described in mathematical symbols or in words,"},{"Start":"00:35.860 ","End":"00:42.815","Text":"and we\u0027ll see plenty of examples and just useful notation to have a general property,"},{"Start":"00:42.815 ","End":"00:48.620","Text":"and I don\u0027t know why I\u0027ve used this funny P. Here\u0027s an example."},{"Start":"00:48.620 ","End":"00:52.740","Text":"For every natural number n,"},{"Start":"00:52.740 ","End":"00:56.495","Text":"5^n minus 1 is divisible by 4."},{"Start":"00:56.495 ","End":"01:00.815","Text":"This is the P of n. For a particular n,"},{"Start":"01:00.815 ","End":"01:05.090","Text":"P of n says 5^n minus 1 is divisible by 4."},{"Start":"01:05.090 ","End":"01:11.810","Text":"For example, P of 1 says that 5^1 minus 1 is divisible by 4."},{"Start":"01:11.810 ","End":"01:16.830","Text":"This happens to be true because 4 divides 4."},{"Start":"01:16.830 ","End":"01:19.745","Text":"Using the symbol, the vertical line means that"},{"Start":"01:19.745 ","End":"01:23.545","Text":"this divides this or this is divisible by this."},{"Start":"01:23.545 ","End":"01:30.050","Text":"P of 2 is the claim that 5 squared minus 1 is divisible by 4."},{"Start":"01:30.050 ","End":"01:33.980","Text":"That\u0027s also true because 25 minus 1 is 24,"},{"Start":"01:33.980 ","End":"01:36.245","Text":"and 24 is divisible by 4."},{"Start":"01:36.245 ","End":"01:38.330","Text":"For short, I\u0027m writing it this way."},{"Start":"01:38.330 ","End":"01:42.860","Text":"P of 3 says 5^3 minus 1 is divisible by 4,"},{"Start":"01:42.860 ","End":"01:48.950","Text":"also true because this is 5 cubed is a 125 minus 1 is a 124,"},{"Start":"01:48.950 ","End":"01:52.010","Text":"and 124 is divisible by 4."},{"Start":"01:52.010 ","End":"01:58.445","Text":"We see that P of n is true for n equals 1, 2, 3."},{"Start":"01:58.445 ","End":"02:05.210","Text":"But we can\u0027t say that P of n is true for all n. We\u0027ve just checked it for 3 values."},{"Start":"02:05.210 ","End":"02:07.790","Text":"It happens to be true for all n,"},{"Start":"02:07.790 ","End":"02:08.960","Text":"but we have improved it."},{"Start":"02:08.960 ","End":"02:12.330","Text":"We\u0027ve just illustrated it for n equals 1, 2, and 3."},{"Start":"02:12.330 ","End":"02:15.950","Text":"We could prove it using proof by induction."},{"Start":"02:15.950 ","End":"02:19.969","Text":"Anyway, let\u0027s get on to the next example of a claim."},{"Start":"02:19.969 ","End":"02:23.975","Text":"For every natural number n,"},{"Start":"02:23.975 ","End":"02:28.820","Text":"2^n is bigger or equal to n squared minus 1."},{"Start":"02:28.820 ","End":"02:31.370","Text":"P of n is the claim,"},{"Start":"02:31.370 ","End":"02:36.410","Text":"the statement that 2^n is bigger or equal to n squared minus 1."},{"Start":"02:36.410 ","End":"02:39.530","Text":"That\u0027s for a general n. P of 1,"},{"Start":"02:39.530 ","End":"02:44.570","Text":"claims that 2^1 is bigger or equal to 1 squared minus 1."},{"Start":"02:44.570 ","End":"02:51.620","Text":"Now this happens to be true because this is 2 and this is 0,"},{"Start":"02:51.620 ","End":"02:54.114","Text":"and 2 is bigger or equal to 0."},{"Start":"02:54.114 ","End":"02:59.780","Text":"P of 2 says 2 squared is bigger or equal to 2 squared minus 1,"},{"Start":"02:59.780 ","End":"03:01.659","Text":"which is also true."},{"Start":"03:01.659 ","End":"03:07.470","Text":"P of 3 says that 2 cubed bigger or equal to 3 squared minus 1, also true."},{"Start":"03:07.470 ","End":"03:09.090","Text":"This is 8,"},{"Start":"03:09.090 ","End":"03:14.070","Text":"this is 9 minus 1 is also 8 and do 1 more. P to the fourth."},{"Start":"03:14.070 ","End":"03:18.395","Text":"Says that 2^4 is bigger or equal to 4 squared minus 1,"},{"Start":"03:18.395 ","End":"03:21.020","Text":"and that\u0027s also true."},{"Start":"03:21.020 ","End":"03:26.780","Text":"Now we\u0027ve shown that P of n is true for n equals 1, 2, 3, and 4."},{"Start":"03:26.780 ","End":"03:31.340","Text":"We haven\u0027t shown that it\u0027s true for all n. It happens to be true."},{"Start":"03:31.340 ","End":"03:33.020","Text":"But we need to prove this."},{"Start":"03:33.020 ","End":"03:35.270","Text":"The way to prove it would be proved by induction,"},{"Start":"03:35.270 ","End":"03:36.530","Text":"which we have yet to learn."},{"Start":"03:36.530 ","End":"03:38.195","Text":"It will be in the next clip."},{"Start":"03:38.195 ","End":"03:41.660","Text":"Now third example of a claim like that."},{"Start":"03:41.660 ","End":"03:44.990","Text":"For every natural number n,"},{"Start":"03:44.990 ","End":"03:53.965","Text":"n squared minus n plus 11 is a prime number that\u0027s P of n. N is a variable."},{"Start":"03:53.965 ","End":"03:57.545","Text":"This is a claim about all natural numbers."},{"Start":"03:57.545 ","End":"04:00.680","Text":"We can also substitute specific natural numbers,"},{"Start":"04:00.680 ","End":"04:06.680","Text":"like P of 1 says that 1 squared minus 1 plus 11 is a prime number."},{"Start":"04:06.680 ","End":"04:12.680","Text":"This happens to be true because it comes out to be 11, which is prime."},{"Start":"04:12.680 ","End":"04:14.435","Text":"P of 2,"},{"Start":"04:14.435 ","End":"04:17.990","Text":"2 squared minus 2 plus 11 is a prime number."},{"Start":"04:17.990 ","End":"04:22.445","Text":"Yeah, that\u0027s true because this is 4 minus 2 plus 11 is 13."},{"Start":"04:22.445 ","End":"04:24.845","Text":"True. P of 3,"},{"Start":"04:24.845 ","End":"04:27.610","Text":"17 is a prime, true."},{"Start":"04:27.610 ","End":"04:30.750","Text":"P of 4, 4 squared is 16,"},{"Start":"04:30.750 ","End":"04:32.940","Text":"minus 4 is 12,"},{"Start":"04:32.940 ","End":"04:35.430","Text":"12 plus 11 is 23 is a prime number."},{"Start":"04:35.430 ","End":"04:37.535","Text":"True. Looks good."},{"Start":"04:37.535 ","End":"04:41.830","Text":"P of 5 comes out to be 31 is a prime number."},{"Start":"04:41.830 ","End":"04:45.910","Text":"True. 41 is a prime number."},{"Start":"04:45.910 ","End":"04:49.650","Text":"True. 53 is a prime number,"},{"Start":"04:49.650 ","End":"04:50.820","Text":"that\u0027s P of 7."},{"Start":"04:50.820 ","End":"04:53.390","Text":"True. 67 is a prime number,"},{"Start":"04:53.390 ","End":"04:55.820","Text":"83 is a prime number,"},{"Start":"04:55.820 ","End":"04:58.040","Text":"10 squared minus 10 plus 11,"},{"Start":"04:58.040 ","End":"04:59.795","Text":"it\u0027s 101 is a prime number."},{"Start":"04:59.795 ","End":"05:02.750","Text":"True. They might think, \u0027\u0027Okay,"},{"Start":"05:02.750 ","End":"05:04.520","Text":"we\u0027ve checked 1, 2,"},{"Start":"05:04.520 ","End":"05:05.870","Text":"3, 4, 5,"},{"Start":"05:05.870 ","End":"05:07.505","Text":"6, 7, 8, 9, 10."},{"Start":"05:07.505 ","End":"05:10.130","Text":"P of n is true for n equals all these.\u0027\u0027"},{"Start":"05:10.130 ","End":"05:13.220","Text":"But it doesn\u0027t prove that it\u0027s true for all n,"},{"Start":"05:13.220 ","End":"05:14.870","Text":"it\u0027s just 10 examples."},{"Start":"05:14.870 ","End":"05:17.030","Text":"As a matter of fact,"},{"Start":"05:17.030 ","End":"05:19.370","Text":"if you took to shoot 11,"},{"Start":"05:19.370 ","End":"05:24.920","Text":"P of 11 says that 11 squared minus 11 plus 11 is a prime number."},{"Start":"05:24.920 ","End":"05:27.140","Text":"Well, minus 11 plus 11 cancels,"},{"Start":"05:27.140 ","End":"05:29.750","Text":"so it\u0027s 11 squared is not a prime number,"},{"Start":"05:29.750 ","End":"05:30.980","Text":"it\u0027s a squares,"},{"Start":"05:30.980 ","End":"05:32.780","Text":"obviously not a prime number, it\u0027s false."},{"Start":"05:32.780 ","End":"05:34.895","Text":"If we just stopped at 10,"},{"Start":"05:34.895 ","End":"05:36.290","Text":"you might think, yeah."},{"Start":"05:36.290 ","End":"05:38.405","Text":"No matter how many you check,"},{"Start":"05:38.405 ","End":"05:40.180","Text":"it\u0027s not a proof."},{"Start":"05:40.180 ","End":"05:42.555","Text":"Just for interest\u0027s sake,"},{"Start":"05:42.555 ","End":"05:45.560","Text":"if you put 41 here,"},{"Start":"05:45.560 ","End":"05:52.070","Text":"it turns out that n squared minus n plus 41 is a prime number for n equals 1 to 40."},{"Start":"05:52.070 ","End":"05:54.470","Text":"You could check 40 examples and think, Oh yeah,"},{"Start":"05:54.470 ","End":"05:58.865","Text":"we found a formula that generates prime numbers and you\u0027d be wrong."},{"Start":"05:58.865 ","End":"06:00.870","Text":"Now, claim Number 4,"},{"Start":"06:00.870 ","End":"06:03.350","Text":"for every natural number n,"},{"Start":"06:03.350 ","End":"06:06.530","Text":"the sum of the squares 1 square plus 2 square plus 3 split"},{"Start":"06:06.530 ","End":"06:10.345","Text":"up to n squared is given by this formula."},{"Start":"06:10.345 ","End":"06:13.980","Text":"P of n this says for"},{"Start":"06:13.980 ","End":"06:18.650","Text":"a particular n. Let\u0027s check some examples what happens when n equals 1."},{"Start":"06:18.650 ","End":"06:21.065","Text":"It says that 1 squared,"},{"Start":"06:21.065 ","End":"06:25.335","Text":"because some stops at 1 squared is only 1 term,"},{"Start":"06:25.335 ","End":"06:27.270","Text":"is equal to 1 sticks of 1,"},{"Start":"06:27.270 ","End":"06:29.445","Text":"1 plus 1 twice 1 plus 1,"},{"Start":"06:29.445 ","End":"06:31.410","Text":"which happens to be true,"},{"Start":"06:31.410 ","End":"06:34.070","Text":"both sides of the equation come out to be 1."},{"Start":"06:34.070 ","End":"06:37.975","Text":"This is 1 times 2 times 3 over 6."},{"Start":"06:37.975 ","End":"06:39.975","Text":"6 over 6 is 1 it\u0027s true."},{"Start":"06:39.975 ","End":"06:42.845","Text":"P of 2, we have 1 squared plus 2 squared."},{"Start":"06:42.845 ","End":"06:45.185","Text":"If you plug 2 here, you get this."},{"Start":"06:45.185 ","End":"06:48.110","Text":"Check both sides, they both come out to be 5."},{"Start":"06:48.110 ","End":"06:49.430","Text":"Again, it\u0027s true."},{"Start":"06:49.430 ","End":"06:52.610","Text":"P of 3 says this equals this."},{"Start":"06:52.610 ","End":"06:54.440","Text":"They both come out to be 14."},{"Start":"06:54.440 ","End":"06:58.685","Text":"True. P of 4 also turns out to be true."},{"Start":"06:58.685 ","End":"07:02.580","Text":"So P of n works for n equals 1,"},{"Start":"07:02.580 ","End":"07:05.185","Text":"2, 3, and 4."},{"Start":"07:05.185 ","End":"07:11.089","Text":"But it doesn\u0027t prove that it\u0027s true for all n. It happens to be true."},{"Start":"07:11.089 ","End":"07:12.650","Text":"It is true for all n,"},{"Start":"07:12.650 ","End":"07:14.480","Text":"but this doesn\u0027t prove it."},{"Start":"07:14.480 ","End":"07:19.505","Text":"In fact, this example will be proven in the following clip."},{"Start":"07:19.505 ","End":"07:22.280","Text":"2 more statements, claims,"},{"Start":"07:22.280 ","End":"07:27.740","Text":"propositions need to define a recursive sequence by a_1 is"},{"Start":"07:27.740 ","End":"07:29.825","Text":"the square root of 2 and"},{"Start":"07:29.825 ","End":"07:34.475","Text":"each subsequent a_n can be gotten from the current a_n by this formula,"},{"Start":"07:34.475 ","End":"07:37.630","Text":"a_n plus 1 is the square root of a_n plus 2."},{"Start":"07:37.630 ","End":"07:39.990","Text":"So if we plug in a_1 we get a_2,"},{"Start":"07:39.990 ","End":"07:41.380","Text":"plug in a_2, we get a_3."},{"Start":"07:41.380 ","End":"07:43.114","Text":"That\u0027s how we build the sequence."},{"Start":"07:43.114 ","End":"07:45.505","Text":"Claim Number 5."},{"Start":"07:45.505 ","End":"07:47.910","Text":"For every natural number n,"},{"Start":"07:47.910 ","End":"07:50.385","Text":"a_n is less than or equal to 2."},{"Start":"07:50.385 ","End":"07:55.070","Text":"Let\u0027s see, P n says that a_n is less than or equal to 2 for"},{"Start":"07:55.070 ","End":"08:00.875","Text":"a particular n. The claim is that P of n is true for every n. Let\u0027s check some values."},{"Start":"08:00.875 ","End":"08:07.545","Text":"A_1 is square root of 2 and square root of 2 is less than or equal to 2."},{"Start":"08:07.545 ","End":"08:10.870","Text":"So P of 1 is true."},{"Start":"08:10.870 ","End":"08:15.630","Text":"A_2 is the square root of a_n plus 2,"},{"Start":"08:15.630 ","End":"08:17.925","Text":"which is a_1 plus 2."},{"Start":"08:17.925 ","End":"08:25.085","Text":"We\u0027ve got a_1 from here and comes out to be something that\u0027s less than or equal to 2."},{"Start":"08:25.085 ","End":"08:28.595","Text":"The statement P of 2 is true."},{"Start":"08:28.595 ","End":"08:35.370","Text":"A_3 is the square root of this plus 2,"},{"Start":"08:35.370 ","End":"08:39.050","Text":"and if you are computing it, you wouldn\u0027t compute it from scratch because you"},{"Start":"08:39.050 ","End":"08:42.710","Text":"would take this part to be a_2."},{"Start":"08:42.710 ","End":"08:43.790","Text":"Whatever that comes out,"},{"Start":"08:43.790 ","End":"08:45.500","Text":"you substitute that in here."},{"Start":"08:45.500 ","End":"08:49.040","Text":"Anyway, does come out to be less than or equal to 2."},{"Start":"08:49.040 ","End":"08:52.385","Text":"We\u0027ve proven P of 3 also."},{"Start":"08:52.385 ","End":"08:59.765","Text":"A_4 is the square root of this part is a_3 and"},{"Start":"08:59.765 ","End":"09:03.380","Text":"square root of a_3 plus 2 comes out to be the square root of 1.96"},{"Start":"09:03.380 ","End":"09:07.250","Text":"plus 2 comes out to be still less than or equal to 2,"},{"Start":"09:07.250 ","End":"09:09.550","Text":"so that verifies P 4."},{"Start":"09:09.550 ","End":"09:14.030","Text":"We\u0027ve shown that P of n is true for n equals 1, 2, 3, and 4."},{"Start":"09:14.030 ","End":"09:19.925","Text":"But it doesn\u0027t prove that P of n is true for all n. This still happen to be true."},{"Start":"09:19.925 ","End":"09:25.040","Text":"It appears in 1 of the exercises following the tutorial clips,"},{"Start":"09:25.040 ","End":"09:28.655","Text":"and continuing with this same recursive sequence."},{"Start":"09:28.655 ","End":"09:31.575","Text":"Last claim, for every natural number n,"},{"Start":"09:31.575 ","End":"09:34.760","Text":"a_n is less than or equal to a _ plus 1."},{"Start":"09:34.760 ","End":"09:37.100","Text":"What it says is that for a particular n,"},{"Start":"09:37.100 ","End":"09:39.380","Text":"a_n less than or equal to a_n plus 1."},{"Start":"09:39.380 ","End":"09:42.740","Text":"Let\u0027s write some values of a_n for n equals 1, 2, 3, and 4."},{"Start":"09:42.740 ","End":"09:44.255","Text":"Just copying them from here,"},{"Start":"09:44.255 ","End":"09:46.315","Text":"here, here, and here."},{"Start":"09:46.315 ","End":"09:50.390","Text":"From here we see that this is less than or equal to this,"},{"Start":"09:50.390 ","End":"09:51.620","Text":"this is less than or equal to this,"},{"Start":"09:51.620 ","End":"09:53.335","Text":"this is less than or equal to this."},{"Start":"09:53.335 ","End":"09:55.770","Text":"From this inequality, P of 1 is true,"},{"Start":"09:55.770 ","End":"09:57.675","Text":"then P of 2 is true,"},{"Start":"09:57.675 ","End":"10:00.150","Text":"and P of 3 is also true."},{"Start":"10:00.150 ","End":"10:03.575","Text":"We\u0027ve proven P of n for n equals 1, 2, 3."},{"Start":"10:03.575 ","End":"10:05.540","Text":"But you\u0027ve got the idea by now."},{"Start":"10:05.540 ","End":"10:08.135","Text":"It matter how many of these you check,"},{"Start":"10:08.135 ","End":"10:14.100","Text":"it doesn\u0027t prove that it\u0027s true for all n. This example happens to be true for all n,"},{"Start":"10:14.100 ","End":"10:16.500","Text":"it\u0027s just that we haven\u0027t proven it here."},{"Start":"10:16.500 ","End":"10:19.490","Text":"That\u0027s enough for this clip."}],"ID":26638},{"Watched":false,"Name":"How to prove using Induction","Duration":"5m 43s","ChapterTopicVideoID":25831,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"This clip is the practical clip which shows how we"},{"Start":"00:04.860 ","End":"00:09.405","Text":"use induction to prove claims about natural numbers."},{"Start":"00:09.405 ","End":"00:11.640","Text":"In the previous clip, we gave an introduction,"},{"Start":"00:11.640 ","End":"00:13.890","Text":"we didn\u0027t show the method itself,"},{"Start":"00:13.890 ","End":"00:15.255","Text":"we\u0027ll do that here,"},{"Start":"00:15.255 ","End":"00:16.290","Text":"and in the following clip,"},{"Start":"00:16.290 ","End":"00:17.790","Text":"we\u0027ll talk about the why,"},{"Start":"00:17.790 ","End":"00:19.710","Text":"of why this method works."},{"Start":"00:19.710 ","End":"00:22.215","Text":"I want to remind you that it\u0027s a prove that\u0027s used for"},{"Start":"00:22.215 ","End":"00:26.640","Text":"natural numbers and it\u0027s a tool that sometimes works."},{"Start":"00:26.640 ","End":"00:30.330","Text":"Wouldn\u0027t say that every claim about natural numbers can be proved by induction,"},{"Start":"00:30.330 ","End":"00:32.820","Text":"usually not, but often it is."},{"Start":"00:32.820 ","End":"00:35.995","Text":"Let\u0027s take the following example,"},{"Start":"00:35.995 ","End":"00:39.410","Text":"and often the question itself says prove by induction."},{"Start":"00:39.410 ","End":"00:41.240","Text":"So you know to use induction."},{"Start":"00:41.240 ","End":"00:45.420","Text":"Prove by induction that for each natural number n,"},{"Start":"00:45.420 ","End":"00:50.975","Text":"1 squared plus 2 squared and so on up to n squared is equal to this expression."},{"Start":"00:50.975 ","End":"00:55.820","Text":"Sometimes this sum will be expressed in Sigma notation."},{"Start":"00:55.820 ","End":"00:59.660","Text":"I don\u0027t know whether you have or haven\u0027t learned Sigma notation,"},{"Start":"00:59.660 ","End":"01:01.085","Text":"so I\u0027ll assume you haven\u0027t."},{"Start":"01:01.085 ","End":"01:04.600","Text":"But in case you have, this is how we would write the left-hand side."},{"Start":"01:04.600 ","End":"01:08.210","Text":"If you do know Sigma notation and you get the question in this form,"},{"Start":"01:08.210 ","End":"01:14.270","Text":"I would recommend switching it to this form with the ellipsis dot-dot-dot."},{"Start":"01:14.270 ","End":"01:17.585","Text":"This form is usually easier with induction."},{"Start":"01:17.585 ","End":"01:20.450","Text":"Anyway, let\u0027s start the solution."},{"Start":"01:20.450 ","End":"01:24.530","Text":"In general, we want the statement to reclaim depending on"},{"Start":"01:24.530 ","End":"01:28.980","Text":"n and that will be the statement that 1 squared plus"},{"Start":"01:28.980 ","End":"01:34.280","Text":"2 squared and so on plus n squared is equal to this expression for"},{"Start":"01:34.280 ","End":"01:41.885","Text":"a particular n. The first step in induction is to check that P of 1 is true,"},{"Start":"01:41.885 ","End":"01:43.640","Text":"you don\u0027t have to say is true."},{"Start":"01:43.640 ","End":"01:47.675","Text":"If I say 4 is less than 5,"},{"Start":"01:47.675 ","End":"01:51.065","Text":"that\u0027s the same as saying that 4 less than 5 is true."},{"Start":"01:51.065 ","End":"01:55.910","Text":"To say that 2 plus 2 equals 4 is to say that 2 plus 2 equals 4 is true."},{"Start":"01:55.910 ","End":"01:58.715","Text":"In general, a statement can be true or false."},{"Start":"01:58.715 ","End":"02:03.050","Text":"Okay. This is called the base case and in our example,"},{"Start":"02:03.050 ","End":"02:05.420","Text":"P of 1, if you plug 1 in here,"},{"Start":"02:05.420 ","End":"02:09.320","Text":"then this whole sum collapses with just 1 term,"},{"Start":"02:09.320 ","End":"02:12.880","Text":"just the 1 squared and here we have 1/6,"},{"Start":"02:12.880 ","End":"02:15.260","Text":"well I put the 6 on the denominator here."},{"Start":"02:15.260 ","End":"02:17.795","Text":"1, 1 plus 1 twice 1 plus 1."},{"Start":"02:17.795 ","End":"02:22.910","Text":"It\u0027s true because it boils down to 1 equals 1 times 2 times 3 over 6,"},{"Start":"02:22.910 ","End":"02:25.030","Text":"which is definitely true."},{"Start":"02:25.030 ","End":"02:29.475","Text":"By the way, if we include 0 as a natural number,"},{"Start":"02:29.475 ","End":"02:32.255","Text":"remember natural numbers sometimes do include 0,"},{"Start":"02:32.255 ","End":"02:33.380","Text":"sometimes they don\u0027t,"},{"Start":"02:33.380 ","End":"02:36.050","Text":"in that case, the base case would be 0."},{"Start":"02:36.050 ","End":"02:38.705","Text":"We would prove P of 0 is true."},{"Start":"02:38.705 ","End":"02:43.115","Text":"The second step is just to suppose,"},{"Start":"02:43.115 ","End":"02:46.490","Text":"assume that the claim P of n is true for"},{"Start":"02:46.490 ","End":"02:50.600","Text":"a particular n. This is called the induction hypothesis."},{"Start":"02:50.600 ","End":"02:53.960","Text":"It looks the same as what I\u0027m saying here in general,"},{"Start":"02:53.960 ","End":"02:57.815","Text":"but here it\u0027s not for all n, it\u0027s for a particular n."},{"Start":"02:57.815 ","End":"03:01.400","Text":"We assume that for some particular but general n,"},{"Start":"03:01.400 ","End":"03:02.795","Text":"that this is true,"},{"Start":"03:02.795 ","End":"03:04.410","Text":"this becomes the given,"},{"Start":"03:04.410 ","End":"03:12.485","Text":"and step 3 is to use this assumption to prove that P n plus 1 is true."},{"Start":"03:12.485 ","End":"03:14.000","Text":"So that in our case,"},{"Start":"03:14.000 ","End":"03:16.830","Text":"we would conclude that 1 squared plus 2 squared"},{"Start":"03:16.830 ","End":"03:20.360","Text":"up to n plus 1 squared is like this,"},{"Start":"03:20.360 ","End":"03:29.030","Text":"but with n plus 1 instead of n. This we assume is true for a particular n,"},{"Start":"03:29.030 ","End":"03:36.410","Text":"and we use this to show that it\u0027s true for the successive n, for n plus 1."},{"Start":"03:36.410 ","End":"03:39.395","Text":"Now, this statement is equivalent to,"},{"Start":"03:39.395 ","End":"03:45.320","Text":"and this is a trick you often use for expression with the ellipsis."},{"Start":"03:45.320 ","End":"03:51.390","Text":"You tried to bring in the induction hypothesis case with n,"},{"Start":"03:51.390 ","End":"03:55.750","Text":"because this is dot-dot-dot before the n plus 1 squared there was an n-squared,"},{"Start":"03:55.750 ","End":"03:58.263","Text":"so we explicitly write this term in"},{"Start":"03:58.263 ","End":"04:01.735","Text":"because then we can replace it by this from here."},{"Start":"04:01.735 ","End":"04:03.350","Text":"This is what we have to prove,"},{"Start":"04:03.350 ","End":"04:05.120","Text":"I also simplified the right-hand side."},{"Start":"04:05.120 ","End":"04:07.010","Text":"Put 6 in the denominator,"},{"Start":"04:07.010 ","End":"04:08.465","Text":"here n plus 2,"},{"Start":"04:08.465 ","End":"04:10.400","Text":"here 2n plus 3."},{"Start":"04:10.400 ","End":"04:14.840","Text":"By the induction hypothesis for our particular n,"},{"Start":"04:14.840 ","End":"04:19.920","Text":"this is equal to this,"},{"Start":"04:19.920 ","End":"04:22.980","Text":"so we can put this here."},{"Start":"04:22.980 ","End":"04:26.105","Text":"This is the expression we\u0027re trying to prove."},{"Start":"04:26.105 ","End":"04:31.145","Text":"Let\u0027s see, this is true if and only if we can divide both sides by n plus 1."},{"Start":"04:31.145 ","End":"04:32.840","Text":"This goes, this goes,"},{"Start":"04:32.840 ","End":"04:35.825","Text":"and power of 2 goes."},{"Start":"04:35.825 ","End":"04:38.600","Text":"I also multiplied by 6."},{"Start":"04:38.600 ","End":"04:42.105","Text":"Yeah, I should say the 6 disappears from here and here"},{"Start":"04:42.105 ","End":"04:45.070","Text":"and we get a 6 in front of this."},{"Start":"04:45.070 ","End":"04:48.190","Text":"This is what we have to prove is equivalent to this."},{"Start":"04:48.190 ","End":"04:52.820","Text":"Then if you expand both sides to get a polynomial,"},{"Start":"04:52.820 ","End":"04:56.350","Text":"a quadratic in n, we get that this equals this."},{"Start":"04:56.350 ","End":"04:58.380","Text":"Left and right sides are the same,"},{"Start":"04:58.380 ","End":"04:59.760","Text":"so this is true."},{"Start":"04:59.760 ","End":"05:02.340","Text":"Because this is true, this is true, therefore this is true,"},{"Start":"05:02.340 ","End":"05:04.050","Text":"and this is true,"},{"Start":"05:04.050 ","End":"05:06.330","Text":"and that concludes this proof."},{"Start":"05:06.330 ","End":"05:10.985","Text":"Just a remark that often we combine steps 2 and 3."},{"Start":"05:10.985 ","End":"05:14.810","Text":"Instead of separately taking given P of n,"},{"Start":"05:14.810 ","End":"05:16.955","Text":"prove P of n plus 1,"},{"Start":"05:16.955 ","End":"05:21.060","Text":"we combine them into an if-then statement."},{"Start":"05:21.060 ","End":"05:24.055","Text":"If P of n, then P of n plus 1."},{"Start":"05:24.055 ","End":"05:26.330","Text":"It\u0027s not circular reasoning."},{"Start":"05:26.330 ","End":"05:30.830","Text":"We\u0027re trying to prove P of n for all n. We assume that it\u0027s"},{"Start":"05:30.830 ","End":"05:34.543","Text":"true for a particular n and show that it\u0027s true for the following n."},{"Start":"05:34.543 ","End":"05:36.650","Text":"Anyway, you\u0027ll get used to it."},{"Start":"05:36.650 ","End":"05:40.860","Text":"We will explain the logic in one of the following clips."}],"ID":26635},{"Watched":false,"Name":"Solved Examples","Duration":"6m 49s","ChapterTopicVideoID":25832,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"Continuing after the break."},{"Start":"00:02.460 ","End":"00:03.990","Text":"Here\u0027s Example 2,"},{"Start":"00:03.990 ","End":"00:06.570","Text":"which is quite similar to Example 1,"},{"Start":"00:06.570 ","End":"00:09.840","Text":"we have to prove by induction that the sum of"},{"Start":"00:09.840 ","End":"00:15.000","Text":"successive cubes from 1 cubed up to n cubed is given by this formula"},{"Start":"00:15.000 ","End":"00:18.690","Text":"a quarter m squared and plus 1 squared for"},{"Start":"00:18.690 ","End":"00:26.145","Text":"all natural numbers from 1 onwards."},{"Start":"00:26.145 ","End":"00:30.540","Text":"I like to use the P notation for short, P of n,"},{"Start":"00:30.540 ","End":"00:38.515","Text":"here will be the statement that 1 cubed plus 2 cubed plus n cubed is equal to this."},{"Start":"00:38.515 ","End":"00:43.460","Text":"We want to prove that P of n is true for all n. We\u0027ll follow the same steps as before."},{"Start":"00:43.460 ","End":"00:45.200","Text":"The first step is the base case."},{"Start":"00:45.200 ","End":"00:47.390","Text":"We\u0027ll check that p of 1 is true,"},{"Start":"00:47.390 ","End":"00:51.060","Text":"so that 1 cubed is equal to 1 quarter,"},{"Start":"00:51.060 ","End":"00:53.340","Text":"1 squared times 1 plus 1 squared,"},{"Start":"00:53.340 ","End":"00:54.830","Text":"and that comes out true,"},{"Start":"00:54.830 ","End":"00:56.570","Text":"both sides are equal to 1."},{"Start":"00:56.570 ","End":"00:58.625","Text":"Here we have 2 squared over 4."},{"Start":"00:58.625 ","End":"01:05.150","Text":"Next step is to just suppose that the claim P of n is true for a particular end,"},{"Start":"01:05.150 ","End":"01:07.775","Text":"the induction hypothesis part."},{"Start":"01:07.775 ","End":"01:09.695","Text":"For this particular end,"},{"Start":"01:09.695 ","End":"01:12.480","Text":"we have the following equality,"},{"Start":"01:12.480 ","End":"01:14.790","Text":"and we take this as the given."},{"Start":"01:14.790 ","End":"01:21.000","Text":"Step 3 is to use this assumption to prove that P of"},{"Start":"01:21.000 ","End":"01:27.365","Text":"n plus 1 is true for the same n i.e that 1 cubed plus 2 cubed,"},{"Start":"01:27.365 ","End":"01:28.730","Text":"so on and so on."},{"Start":"01:28.730 ","End":"01:30.975","Text":"Up to n plus 1 cubed."},{"Start":"01:30.975 ","End":"01:34.580","Text":"Same formula with n plus 1 instead of n. That"},{"Start":"01:34.580 ","End":"01:38.240","Text":"comes out to be equivalent to 2 things here."},{"Start":"01:38.240 ","End":"01:41.780","Text":"First of all, I included the term before last,"},{"Start":"01:41.780 ","End":"01:46.850","Text":"the n cubed because you\u0027re going to use the induction hypothesis on the right,"},{"Start":"01:46.850 ","End":"01:48.500","Text":"just a little bit of algebra."},{"Start":"01:48.500 ","End":"01:52.010","Text":"n plus 1, plus 1 is n plus 2. Ready that all."},{"Start":"01:52.010 ","End":"01:53.860","Text":"We want to prove this,"},{"Start":"01:53.860 ","End":"01:59.660","Text":"we\u0027ll use the induction hypothesis to replace this by this."},{"Start":"02:01.140 ","End":"02:05.425","Text":"This is this, and that\u0027s the rest of it."},{"Start":"02:05.425 ","End":"02:13.120","Text":"Then n plus 1 squared cancels here and here and here it goes from 3 down to 1,"},{"Start":"02:13.120 ","End":"02:15.985","Text":"we\u0027ll also multiply both sides by 4."},{"Start":"02:15.985 ","End":"02:18.080","Text":"There will be a 4 here."},{"Start":"02:18.080 ","End":"02:21.930","Text":"That will give us the following expand,"},{"Start":"02:21.930 ","End":"02:23.880","Text":"and we get on both side,"},{"Start":"02:23.880 ","End":"02:25.965","Text":"n squared plus 4n plus 4."},{"Start":"02:25.965 ","End":"02:27.825","Text":"This is certainly true,"},{"Start":"02:27.825 ","End":"02:33.295","Text":"and that proves what we wanted to show the n plus 1 case."},{"Start":"02:33.295 ","End":"02:35.440","Text":"That completes this example."},{"Start":"02:35.440 ","End":"02:38.840","Text":"Let\u0027s do another. In this example,"},{"Start":"02:38.840 ","End":"02:44.630","Text":"we have to prove by induction that the sum of consecutive natural numbers from 1"},{"Start":"02:44.630 ","End":"02:52.000","Text":"up to 2n is given by the expression n times 2n plus 1."},{"Start":"02:52.000 ","End":"02:59.615","Text":"We\u0027ll let P of n be the statement that this is equal to this first P of 1."},{"Start":"02:59.615 ","End":"03:01.640","Text":"If you plug in n equals 1,"},{"Start":"03:01.640 ","End":"03:04.280","Text":"you want the sum from 1 up to 2."},{"Start":"03:04.280 ","End":"03:07.975","Text":"Here we have 1 times twice 1 plus 1,"},{"Start":"03:07.975 ","End":"03:12.109","Text":"and both sides come out to be 3. That\u0027s fine."},{"Start":"03:12.109 ","End":"03:15.020","Text":"Next, the induction hypothesis P of n is true for"},{"Start":"03:15.020 ","End":"03:21.020","Text":"a particular n. Which means that this is equal to this."},{"Start":"03:21.020 ","End":"03:22.640","Text":"For a particular n,"},{"Start":"03:22.640 ","End":"03:24.365","Text":"this is our given."},{"Start":"03:24.365 ","End":"03:28.700","Text":"Step 3, we\u0027ll use this to prove that P of n"},{"Start":"03:28.700 ","End":"03:32.990","Text":"plus 1 is true for the following n. In other words,"},{"Start":"03:32.990 ","End":"03:38.960","Text":"that 1 plus 2 plus 3 and so on up to twice n plus 1 is equal to this expression."},{"Start":"03:38.960 ","End":"03:42.410","Text":"That\u0027s equivalent to a bit of algebra."},{"Start":"03:42.410 ","End":"03:44.060","Text":"Open this up. Also,"},{"Start":"03:44.060 ","End":"03:47.060","Text":"twice m plus 1 plus 1 is 2n plus 3."},{"Start":"03:47.060 ","End":"03:52.205","Text":"Ow what we can do, we can throw in some extra terms,"},{"Start":"03:52.205 ","End":"03:55.465","Text":"the 1 before last and the 1 before that."},{"Start":"03:55.465 ","End":"03:57.570","Text":"We have up to 2n,"},{"Start":"03:57.570 ","End":"03:58.800","Text":"then 2n plus 1,"},{"Start":"03:58.800 ","End":"04:01.200","Text":"then 2n plus 2."},{"Start":"04:01.200 ","End":"04:03.485","Text":"On the right, the same thing."},{"Start":"04:03.485 ","End":"04:07.685","Text":"This is what we have to prove given this."},{"Start":"04:07.685 ","End":"04:10.370","Text":"All we have to do is substitute."},{"Start":"04:10.370 ","End":"04:14.030","Text":"Instead of this, we can put n times 2n plus 1,"},{"Start":"04:14.030 ","End":"04:16.280","Text":"and the rest of it just as is."},{"Start":"04:16.280 ","End":"04:18.245","Text":"We have to prove this."},{"Start":"04:18.245 ","End":"04:23.360","Text":"This is the same as just open up the brackets with"},{"Start":"04:23.360 ","End":"04:28.040","Text":"algebra and then simplify a bit like here,"},{"Start":"04:28.040 ","End":"04:29.990","Text":"n plus 2n plus 2n is 5n."},{"Start":"04:29.990 ","End":"04:31.430","Text":"I\u0027ll leave you to check the algebra."},{"Start":"04:31.430 ","End":"04:32.750","Text":"We have that this equals this,"},{"Start":"04:32.750 ","End":"04:34.414","Text":"which is certainly true."},{"Start":"04:34.414 ","End":"04:36.560","Text":"That concludes this example."},{"Start":"04:36.560 ","End":"04:38.420","Text":"Let\u0027s do another one."},{"Start":"04:38.420 ","End":"04:41.510","Text":"This time a divisibility problem."},{"Start":"04:41.510 ","End":"04:47.990","Text":"Prove by induction that 10 to the n minus 1 is divisible by 9 for each natural number n,"},{"Start":"04:47.990 ","End":"04:53.495","Text":"I could write a shorthand for all n in set of natural numbers."},{"Start":"04:53.495 ","End":"04:57.470","Text":"9 divides 10 to the n minus 1."},{"Start":"04:57.470 ","End":"05:01.250","Text":"I\u0027ll denote P of n to be the statement 9"},{"Start":"05:01.250 ","End":"05:05.875","Text":"divides 10 to the n minus 1 or 10 to the n minus 1 is divisible by 9."},{"Start":"05:05.875 ","End":"05:07.695","Text":"Check P of 1."},{"Start":"05:07.695 ","End":"05:12.380","Text":"When n is 1, we get 10 to the 1 minus 1 is 9,"},{"Start":"05:12.380 ","End":"05:14.675","Text":"and certainly 9 divides 9."},{"Start":"05:14.675 ","End":"05:16.850","Text":"Then the induction hypothesis,"},{"Start":"05:16.850 ","End":"05:21.020","Text":"we\u0027ll assume that P of n is true for a particular n. In other words,"},{"Start":"05:21.020 ","End":"05:25.080","Text":"for this n, 10 to the n minus 1 is divisible by 9."},{"Start":"05:25.080 ","End":"05:28.635","Text":"We want to prove P of n plus 1."},{"Start":"05:28.635 ","End":"05:34.200","Text":"In other words, that 9 divides 10 to the n plus 1 minus 1."},{"Start":"05:34.200 ","End":"05:37.160","Text":"We\u0027ll use this to prove this."},{"Start":"05:37.160 ","End":"05:41.705","Text":"Well, this is true if and only if this is true,"},{"Start":"05:41.705 ","End":"05:45.110","Text":"break up the 10 to the n plus 1 as 10 times 10 to"},{"Start":"05:45.110 ","End":"05:50.895","Text":"the n. Then we can break the 10 up into 9 plus 1."},{"Start":"05:50.895 ","End":"05:52.550","Text":"Expand the brackets."},{"Start":"05:52.550 ","End":"05:56.465","Text":"We have 9 times 10 to the n plus 10 to the n minus 1."},{"Start":"05:56.465 ","End":"06:03.180","Text":"Now, 9 certainly divides 9 times 10 to the n. By the induction hypothesis,"},{"Start":"06:03.180 ","End":"06:05.910","Text":"9 divides 10 to the n minus 1."},{"Start":"06:05.910 ","End":"06:08.375","Text":"If 9 divides both this and this,"},{"Start":"06:08.375 ","End":"06:10.250","Text":"then it divides their sum."},{"Start":"06:10.250 ","End":"06:12.290","Text":"That concludes this proof."},{"Start":"06:12.290 ","End":"06:15.560","Text":"Just like to remark that in this particular case,"},{"Start":"06:15.560 ","End":"06:18.035","Text":"we didn\u0027t need induction to prove it."},{"Start":"06:18.035 ","End":"06:22.295","Text":"I\u0027ll show you why 10 to the n minus 1 is divisible by 9 without induction,"},{"Start":"06:22.295 ","End":"06:24.290","Text":"for example, take n equals 6,"},{"Start":"06:24.290 ","End":"06:26.450","Text":"then 10 to the 6 is a 1,000,000,"},{"Start":"06:26.450 ","End":"06:30.280","Text":"1,000,000 minus 1 is 999,999,"},{"Start":"06:30.280 ","End":"06:32.505","Text":"that\u0027s clearly divisible by 9."},{"Start":"06:32.505 ","End":"06:36.875","Text":"Any natural number other than 6 is the same thing."},{"Start":"06:36.875 ","End":"06:40.835","Text":"We just get 99999 in a row n times,"},{"Start":"06:40.835 ","End":"06:43.055","Text":"that\u0027s clearly divisible by 9."},{"Start":"06:43.055 ","End":"06:44.930","Text":"But that\u0027s just a by the way,"},{"Start":"06:44.930 ","End":"06:48.440","Text":"that sometimes you can find a shorter proof that induction."},{"Start":"06:48.440 ","End":"06:50.700","Text":"That\u0027s it for this clip."}],"ID":26636},{"Watched":false,"Name":"The idea behind Induction","Duration":"5m 49s","ChapterTopicVideoID":25833,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"We\u0027ve learnt the how but not the why of proof by induction."},{"Start":"00:05.385 ","End":"00:11.550","Text":"Let\u0027s summarize the method and then explain what\u0027s the idea behind it, why it works."},{"Start":"00:11.550 ","End":"00:16.260","Text":"Our task is to prove that some property P of n is true for"},{"Start":"00:16.260 ","End":"00:21.915","Text":"all natural numbers n. Here\u0027s how we go about proving it by induction."},{"Start":"00:21.915 ","End":"00:24.480","Text":"Not all tasks can be proved by induction."},{"Start":"00:24.480 ","End":"00:25.770","Text":"But assuming it does,"},{"Start":"00:25.770 ","End":"00:27.480","Text":"that summarize the method."},{"Start":"00:27.480 ","End":"00:31.140","Text":"It consists of 2 or 3 steps depending on how you count."},{"Start":"00:31.140 ","End":"00:35.340","Text":"First of all, we prove the base case that P of 1 is true."},{"Start":"00:35.340 ","End":"00:38.760","Text":"If you count natural numbers including 0,"},{"Start":"00:38.760 ","End":"00:41.415","Text":"then we start with the base case of 0."},{"Start":"00:41.415 ","End":"00:47.720","Text":"Then we prove that given any particular P of n,"},{"Start":"00:47.720 ","End":"00:50.705","Text":"P of n plus 1 is necessarily true,"},{"Start":"00:50.705 ","End":"00:52.595","Text":"that this implies this."},{"Start":"00:52.595 ","End":"00:56.480","Text":"Then the statement will be true by induction."},{"Start":"00:56.480 ","End":"00:59.420","Text":"This step is broken up sometimes into 2 steps."},{"Start":"00:59.420 ","End":"01:03.335","Text":"Take the given as 1 step and then to prove as a separate step,"},{"Start":"01:03.335 ","End":"01:06.920","Text":"which I think I did in the tutorial but here I\u0027ll just"},{"Start":"01:06.920 ","End":"01:11.305","Text":"consider it as 1 step to show that if this, then this."},{"Start":"01:11.305 ","End":"01:15.150","Text":"Now, what\u0027s the idea behind it?"},{"Start":"01:15.150 ","End":"01:16.540","Text":"P of 1,"},{"Start":"01:16.540 ","End":"01:20.390","Text":"of course is true because we proved that directly in the base case."},{"Start":"01:20.390 ","End":"01:23.065","Text":"Now, let\u0027s see why P of 2 is true."},{"Start":"01:23.065 ","End":"01:25.845","Text":"We said that for all n,"},{"Start":"01:25.845 ","End":"01:27.360","Text":"if P of n is true,"},{"Start":"01:27.360 ","End":"01:29.285","Text":"then P of n plus 1 is true."},{"Start":"01:29.285 ","End":"01:32.285","Text":"If you substitute here n equals 1,"},{"Start":"01:32.285 ","End":"01:35.500","Text":"then P of 1 implies P of 2."},{"Start":"01:35.500 ","End":"01:37.890","Text":"But P of 1 is true."},{"Start":"01:37.890 ","End":"01:39.545","Text":"If P of 1 is true,"},{"Start":"01:39.545 ","End":"01:41.390","Text":"and this implies this is true,"},{"Start":"01:41.390 ","End":"01:44.345","Text":"then we can deduce that P of 2 is true."},{"Start":"01:44.345 ","End":"01:47.870","Text":"This is a logical thing called Modus Ponens."},{"Start":"01:47.870 ","End":"01:50.780","Text":"You don\u0027t have to know that there\u0027s a rule in logic that if P"},{"Start":"01:50.780 ","End":"01:53.915","Text":"is true and P implies Q is true,"},{"Start":"01:53.915 ","End":"01:55.505","Text":"then Q is true."},{"Start":"01:55.505 ","End":"01:59.185","Text":"Continuing like this, we know that P of 2 is true,"},{"Start":"01:59.185 ","End":"02:05.090","Text":"and P of 2 implies P of 3 is true from this where we substitute n equals 2."},{"Start":"02:05.090 ","End":"02:08.470","Text":"From this we did use that P of 3."},{"Start":"02:08.470 ","End":"02:11.940","Text":"Since P of 3 and P of 3 implies P of 4,"},{"Start":"02:11.940 ","End":"02:14.085","Text":"we did use P of 4."},{"Start":"02:14.085 ","End":"02:16.445","Text":"If you continue this way,"},{"Start":"02:16.445 ","End":"02:20.690","Text":"you will eventually reach any particular n. P of n is true"},{"Start":"02:20.690 ","End":"02:25.160","Text":"for all n. That\u0027s the logic behind proof by induction."},{"Start":"02:25.160 ","End":"02:30.395","Text":"Now remark, sometimes we don\u0027t start from n equals 1."},{"Start":"02:30.395 ","End":"02:32.210","Text":"Sometimes the claim is made about"},{"Start":"02:32.210 ","End":"02:36.995","Text":"natural numbers that it\u0027s true from some n equals k onwards,"},{"Start":"02:36.995 ","End":"02:40.535","Text":"ie, for n bigger or equal to k. In that case,"},{"Start":"02:40.535 ","End":"02:43.265","Text":"you modify the induction."},{"Start":"02:43.265 ","End":"02:47.370","Text":"We take the base case is n equals k rather than n equals 1."},{"Start":"02:47.370 ","End":"02:49.325","Text":"You prove P of k,"},{"Start":"02:49.325 ","End":"02:50.810","Text":"and the second step is the same."},{"Start":"02:50.810 ","End":"02:52.835","Text":"You prove that if P of n,"},{"Start":"02:52.835 ","End":"03:00.650","Text":"then P of n plus 1 for all natural n bigger or equal to k. An example of this would be"},{"Start":"03:00.650 ","End":"03:04.670","Text":"to show that the inequality 2 to the n is bigger or equal to n squared"},{"Start":"03:04.670 ","End":"03:09.020","Text":"holds for all natural numbers and bigger or equal to 4."},{"Start":"03:09.020 ","End":"03:12.470","Text":"I believe this is in 1 of the exercises that we proved that P of"},{"Start":"03:12.470 ","End":"03:15.950","Text":"4 is true and we still prove that P of n implies P of n plus 1."},{"Start":"03:15.950 ","End":"03:18.455","Text":"Then it\u0027s true for all n bigger or equal to 4."},{"Start":"03:18.455 ","End":"03:20.570","Text":"That\u0027s the 1 remark,"},{"Start":"03:20.570 ","End":"03:23.690","Text":"another remark is that sometimes the claim is"},{"Start":"03:23.690 ","End":"03:26.840","Text":"made not about all natural numbers, but a subset."},{"Start":"03:26.840 ","End":"03:32.360","Text":"Typically for all odd numbers or for all even numbers."},{"Start":"03:32.360 ","End":"03:35.405","Text":"We can modify induction,"},{"Start":"03:35.405 ","End":"03:39.170","Text":"let\u0027s say the case of the odd numbers by proving that P of 1 is true,"},{"Start":"03:39.170 ","End":"03:41.090","Text":"the first odd number."},{"Start":"03:41.090 ","End":"03:44.315","Text":"Then to prove that if P of n is true,"},{"Start":"03:44.315 ","End":"03:50.820","Text":"then P of n plus 2 is true for all odd n. This will show us that if P of 1 is true,"},{"Start":"03:50.820 ","End":"03:53.880","Text":"then P of 3, if P of 3, then P of 5,"},{"Start":"03:53.880 ","End":"03:56.190","Text":"if P of 5, then P of 7 and P of 9,"},{"Start":"03:56.190 ","End":"03:57.570","Text":"P of 11 and so on."},{"Start":"03:57.570 ","End":"03:59.655","Text":"That will be true for all odd numbers."},{"Start":"03:59.655 ","End":"04:05.380","Text":"For example, the claim is that 2 to the n plus 1 is divisible by"},{"Start":"04:05.380 ","End":"04:11.440","Text":"3 for all odd n. Another example this time with the even numbers,"},{"Start":"04:11.440 ","End":"04:16.240","Text":"is that 4^n minus 1 is divisible by 15 for"},{"Start":"04:16.240 ","End":"04:21.990","Text":"all even n. I think this is in 1 of the exercises also."},{"Start":"04:21.990 ","End":"04:27.385","Text":"I\u0027ll conclude this clip with a couple of metaphors for proof by induction."},{"Start":"04:27.385 ","End":"04:29.365","Text":"I got these from the internet."},{"Start":"04:29.365 ","End":"04:34.690","Text":"The first metaphor is that of an infinite ladder."},{"Start":"04:34.730 ","End":"04:37.480","Text":"We suppose you have an infinite ladder."},{"Start":"04:37.480 ","End":"04:41.085","Text":"We\u0027re given that you can reach the first rung of the ladder,"},{"Start":"04:41.085 ","End":"04:46.060","Text":"we\u0027re also given that if we can reach particular rung of the ladder,"},{"Start":"04:46.060 ","End":"04:50.390","Text":"then we can always reach the next rung just simply by climbing up a step."},{"Start":"04:50.390 ","End":"04:52.480","Text":"We can reach Step 1."},{"Start":"04:52.480 ","End":"04:54.065","Text":"If you can reach Step k,"},{"Start":"04:54.065 ","End":"04:56.355","Text":"we can reach Step k plus 1."},{"Start":"04:56.355 ","End":"04:57.700","Text":"It\u0027s clear if that\u0027s the case,"},{"Start":"04:57.700 ","End":"05:01.640","Text":"then we can reach any particular step we want."},{"Start":"05:01.640 ","End":"05:06.820","Text":"This is an example of proof by mathematical induction motivated."},{"Start":"05:06.820 ","End":"05:08.410","Text":"It\u0027s an analogy. It\u0027s not a proof,"},{"Start":"05:08.410 ","End":"05:11.350","Text":"but it helps perhaps understand it."},{"Start":"05:11.350 ","End":"05:14.435","Text":"Now, I\u0027ll give another 1 involving dominoes."},{"Start":"05:14.435 ","End":"05:20.980","Text":"The metaphor with a dominoes is that we have an infinite row of dominoes numbered 1,"},{"Start":"05:20.980 ","End":"05:22.990","Text":"2, 3, 4, and so on."},{"Start":"05:22.990 ","End":"05:26.360","Text":"We\u0027re given that the first domino can be knocked down."},{"Start":"05:26.360 ","End":"05:33.620","Text":"Also that when domino n is knocked down it also knocks down domino n plus 1 with it."},{"Start":"05:33.620 ","End":"05:40.225","Text":"The conclusion is that eventually any domino will be knocked down."},{"Start":"05:40.225 ","End":"05:45.470","Text":"These analogies, the ladder and the domino may help, may not."},{"Start":"05:45.470 ","End":"05:49.620","Text":"With any rate, that\u0027s the end of this clip."}],"ID":26637},{"Watched":false,"Name":"Exercise 1","Duration":"2m 45s","ChapterTopicVideoID":25820,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"In this exercise, we have to prove,"},{"Start":"00:02.550 ","End":"00:05.250","Text":"and I should\u0027ve written by induction,"},{"Start":"00:05.250 ","End":"00:13.980","Text":"that 4 times 10 to the n plus 14 times 19 to the n is divisible by 9 for all n in,"},{"Start":"00:13.980 ","End":"00:18.045","Text":"this is the symbol for natural numbers including 0."},{"Start":"00:18.045 ","End":"00:21.720","Text":"This is a way of making sure that 0 is definitely in."},{"Start":"00:21.720 ","End":"00:25.850","Text":"We\u0027ll let P of n be the property of n that"},{"Start":"00:25.850 ","End":"00:29.795","Text":"4 times 10 to the n plus 14 times 19 to the n is divisible by 9,"},{"Start":"00:29.795 ","End":"00:34.255","Text":"and we have to prove that P of n is true for all n. We,"},{"Start":"00:34.255 ","End":"00:35.910","Text":"first of all, check the base case, which,"},{"Start":"00:35.910 ","End":"00:38.795","Text":"in this case, is n equals 0."},{"Start":"00:38.795 ","End":"00:40.540","Text":"What does that say?"},{"Start":"00:40.540 ","End":"00:43.680","Text":"4 times 10 to the 0 plus 14 times 19 to the 0."},{"Start":"00:43.680 ","End":"00:46.130","Text":"Well, 10 to the 0 is 1, so is 19 to the 0."},{"Start":"00:46.130 ","End":"00:47.885","Text":"4 plus 14 is 18,"},{"Start":"00:47.885 ","End":"00:51.745","Text":"which is divisible by 9, so we\u0027re okay."},{"Start":"00:51.745 ","End":"00:57.545","Text":"Next, we assume that P of n is true for some particular n. For this n,"},{"Start":"00:57.545 ","End":"01:02.020","Text":"4 times 10 to the n plus 14 times 19 to the n is divisible by 9,"},{"Start":"01:02.020 ","End":"01:06.165","Text":"and what we have to do is to prove that for this same n,"},{"Start":"01:06.165 ","End":"01:09.125","Text":"P to the n plus 1 is also true."},{"Start":"01:09.125 ","End":"01:13.400","Text":"In other words, we have to prove that 4 times 10 to the n plus 1 plus 14"},{"Start":"01:13.400 ","End":"01:18.935","Text":"times 19 to the power of n plus 1 is divisible by 9."},{"Start":"01:18.935 ","End":"01:22.530","Text":"Let\u0027s see. We can expand the exponent here."},{"Start":"01:22.530 ","End":"01:24.470","Text":"This is 10 times 10 to the n,"},{"Start":"01:24.470 ","End":"01:28.190","Text":"and this is 19 times 19 to the n. Next,"},{"Start":"01:28.190 ","End":"01:31.530","Text":"we can say 4 times 10 is 40,"},{"Start":"01:31.530 ","End":"01:34.310","Text":"and 14 times 19 is 266."},{"Start":"01:34.310 ","End":"01:36.440","Text":"Use a calculator if you need."},{"Start":"01:36.440 ","End":"01:44.660","Text":"Then we can split this 40 up as 4 plus 36 and this as 14 plus 252."},{"Start":"01:44.660 ","End":"01:48.650","Text":"The way I got this, I was thinking that 10 is 1 plus 9,"},{"Start":"01:48.650 ","End":"01:51.485","Text":"so I did 4 times 1 plus 4 times 9, and here,"},{"Start":"01:51.485 ","End":"01:55.750","Text":"14 times 1, 14 times 18."},{"Start":"01:55.750 ","End":"02:02.745","Text":"Now, expand 4 plus 36 and the 14 plus 252 with the distributive law,"},{"Start":"02:02.745 ","End":"02:06.710","Text":"and then collect together this with this and this"},{"Start":"02:06.710 ","End":"02:10.760","Text":"with this because these terms are divisible by 9."},{"Start":"02:10.760 ","End":"02:12.590","Text":"36 is 9 times 4,"},{"Start":"02:12.590 ","End":"02:16.100","Text":"and 252 is 9 times 28."},{"Start":"02:16.100 ","End":"02:21.830","Text":"This part here is divisible by 9 because of the induction hypothesis."},{"Start":"02:21.830 ","End":"02:29.255","Text":"That\u0027s what we assumed for n. The other half is divisible by 9."},{"Start":"02:29.255 ","End":"02:32.510","Text":"Well, as we saw, it\u0027s got 9 as a factor, and when you add it,"},{"Start":"02:32.510 ","End":"02:34.340","Text":"it\u0027s still divisible by 9,"},{"Start":"02:34.340 ","End":"02:36.545","Text":"so altogether their sum,"},{"Start":"02:36.545 ","End":"02:37.805","Text":"which is this,"},{"Start":"02:37.805 ","End":"02:41.885","Text":"is divisible by 9 proving P of n plus 1,"},{"Start":"02:41.885 ","End":"02:45.720","Text":"and that concludes the proof. We\u0027re done."}],"ID":26624},{"Watched":false,"Name":"Exercise 2","Duration":"4m 2s","ChapterTopicVideoID":25821,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"This exercise is a proof that will do it by induction,"},{"Start":"00:04.335 ","End":"00:07.380","Text":"but it also involves quite a bit of trigonometry."},{"Start":"00:07.380 ","End":"00:09.930","Text":"If you\u0027re not so good with trigonometry,"},{"Start":"00:09.930 ","End":"00:12.000","Text":"you can skip this exercise."},{"Start":"00:12.000 ","End":"00:13.305","Text":"What we have to prove,"},{"Start":"00:13.305 ","End":"00:15.450","Text":"it\u0027s what\u0027s written here."},{"Start":"00:15.450 ","End":"00:19.920","Text":"Like I said, we\u0027ll do it by induction and will appear then be"},{"Start":"00:19.920 ","End":"00:24.119","Text":"the property of n that this is equal to this."},{"Start":"00:24.119 ","End":"00:26.460","Text":"We\u0027ll start with the steps."},{"Start":"00:26.460 ","End":"00:28.230","Text":"First step is the base case."},{"Start":"00:28.230 ","End":"00:30.615","Text":"We\u0027ll check that P of 1 is true."},{"Start":"00:30.615 ","End":"00:34.440","Text":"We\u0027re working with the natural numbers from 1 up."},{"Start":"00:34.440 ","End":"00:37.890","Text":"This says, it\u0027s replacing n with 1,"},{"Start":"00:37.890 ","End":"00:39.290","Text":"we only have 1 term here,"},{"Start":"00:39.290 ","End":"00:41.670","Text":"sine x, and here we have n equals 1,"},{"Start":"00:41.670 ","End":"00:44.510","Text":"1 plus 1 over 2 is 1."},{"Start":"00:44.510 ","End":"00:47.875","Text":"Anyway, we get this. Is this true?"},{"Start":"00:47.875 ","End":"00:54.580","Text":"Yes, because sine 1/2x cancels with sine 1/2x and we\u0027re left with sine x equals sine x,"},{"Start":"00:54.580 ","End":"00:56.765","Text":"so this is true."},{"Start":"00:56.765 ","End":"00:59.480","Text":"Now we\u0027ll assume that for a particular n,"},{"Start":"00:59.480 ","End":"01:00.845","Text":"P of n is true."},{"Start":"01:00.845 ","End":"01:03.260","Text":"This is the induction hypothesis."},{"Start":"01:03.260 ","End":"01:09.170","Text":"This is what it is for this particular n. What we have to show is that for the same n,"},{"Start":"01:09.170 ","End":"01:12.580","Text":"p of n plus 1 is also true."},{"Start":"01:12.580 ","End":"01:16.100","Text":"To spell it out, replace n with n plus 1."},{"Start":"01:16.100 ","End":"01:18.215","Text":"This is what we have to prove,"},{"Start":"01:18.215 ","End":"01:21.240","Text":"and we are given that this is true."},{"Start":"01:21.370 ","End":"01:25.940","Text":"In other words, what we have to show if I just add"},{"Start":"01:25.940 ","End":"01:30.200","Text":"1 more term here before the end of the dot,"},{"Start":"01:30.200 ","End":"01:32.750","Text":"dot, dot, we put a sine and x here."},{"Start":"01:32.750 ","End":"01:35.030","Text":"This is equal to this."},{"Start":"01:35.030 ","End":"01:41.750","Text":"Now we can replace this sum using the induction hypothesis with this."},{"Start":"01:41.750 ","End":"01:46.790","Text":"What we have to prove is that this plus this equals this."},{"Start":"01:46.790 ","End":"01:50.165","Text":"That\u0027s mainly an exercise in trigonometry."},{"Start":"01:50.165 ","End":"01:53.420","Text":"There\u0027s no longer anymore just induction part."},{"Start":"01:53.420 ","End":"01:56.675","Text":"If you wanted to skip the trigonometry, feel free."},{"Start":"01:56.675 ","End":"01:59.210","Text":"For those who want to see it, let\u0027s continue."},{"Start":"01:59.210 ","End":"02:02.000","Text":"This is our trigonometry exercise."},{"Start":"02:02.000 ","End":"02:07.340","Text":"Now look, we have a sine n plus 1 over 2x here and a sine n plus 1 over 2x here."},{"Start":"02:07.340 ","End":"02:11.500","Text":"Let\u0027s take that outside the brackets in this term and"},{"Start":"02:11.500 ","End":"02:16.340","Text":"this term and also we\u0027ll multiply everything by this denominator here."},{"Start":"02:16.340 ","End":"02:21.860","Text":"It disappears from here and here and appears as sine x over 2 here."},{"Start":"02:21.860 ","End":"02:27.260","Text":"Next, bring this term over to the left-hand side with a minus,"},{"Start":"02:27.260 ","End":"02:30.590","Text":"and then we can take it outside the brackets and get sine n plus"},{"Start":"02:30.590 ","End":"02:37.100","Text":"1 over 2x times sine of n over 2x minus this bit here,"},{"Start":"02:37.100 ","End":"02:43.090","Text":"sine n plus 2 over 2x plus this term equals 0."},{"Start":"02:43.090 ","End":"02:48.980","Text":"Now for this, we\u0027ll use the trigonometric identity for sine Alpha minus sine Beta,"},{"Start":"02:48.980 ","End":"02:50.580","Text":"which is this,"},{"Start":"02:50.580 ","End":"02:54.624","Text":"and so expanding this according to this will get this."},{"Start":"02:54.624 ","End":"02:57.400","Text":"Now we\u0027ll simplify this."},{"Start":"02:57.400 ","End":"03:04.320","Text":"n over 2 minus n plus 2 over 2 is just minus 1."},{"Start":"03:04.320 ","End":"03:07.845","Text":"That gives us the minus 1x over 2,"},{"Start":"03:07.845 ","End":"03:12.600","Text":"and n over 2 plus n plus 2 over 2 is 2n plus 2 over 2,"},{"Start":"03:12.600 ","End":"03:14.400","Text":"which is n plus 1."},{"Start":"03:14.400 ","End":"03:17.335","Text":"Here we have n plus 1x over 2."},{"Start":"03:17.335 ","End":"03:19.120","Text":"Now for the parts I\u0027ve colored,"},{"Start":"03:19.120 ","End":"03:22.270","Text":"we can use a trigonometric formula for 2 sine Alpha."},{"Start":"03:22.270 ","End":"03:26.340","Text":"cosine Alpha is equal to sine of 2 Alpha,"},{"Start":"03:26.340 ","End":"03:33.140","Text":"and also here we\u0027ll bring the minus in front because sine is an odd function."},{"Start":"03:33.140 ","End":"03:36.290","Text":"What we end up with is,"},{"Start":"03:36.290 ","End":"03:38.675","Text":"the minus comes out in front,"},{"Start":"03:38.675 ","End":"03:41.465","Text":"then we have the sine x over 2,"},{"Start":"03:41.465 ","End":"03:46.850","Text":"and from these parts will get the sine of twice this,"},{"Start":"03:46.850 ","End":"03:49.220","Text":"which is sine of n plus 1x,"},{"Start":"03:49.220 ","End":"03:52.790","Text":"this part we just copied equals 0."},{"Start":"03:52.790 ","End":"04:00.620","Text":"This is true because we have a minus something plus the same something is 0 finally,"},{"Start":"04:00.620 ","End":"04:03.150","Text":"and this completes the proof."}],"ID":26625},{"Watched":false,"Name":"Exercise 3","Duration":"3m 10s","ChapterTopicVideoID":25822,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this exercise, we\u0027re looking for"},{"Start":"00:02.460 ","End":"00:09.675","Text":"the smallest positive integer k such that from that k onwards,"},{"Start":"00:09.675 ","End":"00:13.170","Text":"2 to the n is bigger or equal to n squared."},{"Start":"00:13.170 ","End":"00:14.535","Text":"Turns out that it exists,"},{"Start":"00:14.535 ","End":"00:16.800","Text":"going to make an intelligent guess and then prove"},{"Start":"00:16.800 ","End":"00:19.740","Text":"that guess by induction. You\u0027ll see what I mean."},{"Start":"00:19.740 ","End":"00:24.615","Text":"That P of n denote 2^n bigger or equal to n squared."},{"Start":"00:24.615 ","End":"00:28.110","Text":"We\u0027re going to prove that P of n is true for all,"},{"Start":"00:28.110 ","End":"00:29.400","Text":"n bigger or equal to k. First,"},{"Start":"00:29.400 ","End":"00:32.250","Text":"we have to find k. Let\u0027s just plug in a few values."},{"Start":"00:32.250 ","End":"00:36.600","Text":"P of 1 says that 2^1 bigger or equal to 1 squared."},{"Start":"00:36.600 ","End":"00:38.205","Text":"Yeah, that\u0027s true."},{"Start":"00:38.205 ","End":"00:41.505","Text":"P of 2, 2 squared bigger or equal to 2 squared."},{"Start":"00:41.505 ","End":"00:44.280","Text":"Yeah. P of 3 is not true,"},{"Start":"00:44.280 ","End":"00:46.635","Text":"2 cubed is 8, 3 squared is 9,"},{"Start":"00:46.635 ","End":"00:49.410","Text":"8 is not bigger or equal to 9."},{"Start":"00:49.410 ","End":"00:51.465","Text":"P of 4, yeah,"},{"Start":"00:51.465 ","End":"00:53.730","Text":"2^4 bigger or equal to 4 squared,"},{"Start":"00:53.730 ","End":"00:55.575","Text":"16 to the 16."},{"Start":"00:55.575 ","End":"00:59.405","Text":"From 4 onwards, it\u0027s going to look like the gap is getting larger."},{"Start":"00:59.405 ","End":"01:02.675","Text":"Here, we have 32 bigger or equal to 25."},{"Start":"01:02.675 ","End":"01:04.009","Text":"We\u0027re going to guess,"},{"Start":"01:04.009 ","End":"01:05.465","Text":"that\u0027s what I mean by estimate,"},{"Start":"01:05.465 ","End":"01:08.475","Text":"that k is equal to 4."},{"Start":"01:08.475 ","End":"01:09.780","Text":"From this point onwards,"},{"Start":"01:09.780 ","End":"01:11.390","Text":"we get only check marks."},{"Start":"01:11.390 ","End":"01:13.720","Text":"Now, we just need to prove it."},{"Start":"01:13.720 ","End":"01:19.125","Text":"What we\u0027re going to prove is that for all n bigger or equal to 4,"},{"Start":"01:19.125 ","End":"01:22.810","Text":"2^n is bigger or equal to n squared."},{"Start":"01:22.810 ","End":"01:27.375","Text":"The base case will be when n is equal to 4,"},{"Start":"01:27.375 ","End":"01:28.995","Text":"and we\u0027ve already done that,"},{"Start":"01:28.995 ","End":"01:30.690","Text":"that\u0027s this here,"},{"Start":"01:30.690 ","End":"01:33.930","Text":"so we just need the induction step."},{"Start":"01:33.930 ","End":"01:36.845","Text":"We assume that P of n is true,"},{"Start":"01:36.845 ","End":"01:41.140","Text":"this is so for some n bigger or equal to 4."},{"Start":"01:41.140 ","End":"01:48.020","Text":"From this, we have to derive that P of n plus 1 is true for this n. In other words,"},{"Start":"01:48.020 ","End":"01:49.610","Text":"that 2^n plus 1,"},{"Start":"01:49.610 ","End":"01:52.235","Text":"bigger or equal to n plus 1 squared."},{"Start":"01:52.235 ","End":"01:54.710","Text":"We use this to prove this."},{"Start":"01:54.710 ","End":"01:57.960","Text":"Let\u0027s see, 2^n bigger or equal to n plus 1 squared,"},{"Start":"01:57.960 ","End":"02:01.430","Text":"let\u0027s see if we can derive this from something that\u0027s definitely true."},{"Start":"02:01.430 ","End":"02:06.155","Text":"Well, this is the same as 2 times 2^n bigger or equal to n plus 1 squared."},{"Start":"02:06.155 ","End":"02:11.790","Text":"This derives from this because we know that 2^n is bigger or equal to n squared."},{"Start":"02:11.790 ","End":"02:13.880","Text":"If we can prove that this is true,"},{"Start":"02:13.880 ","End":"02:16.025","Text":"then it will follow that this is true."},{"Start":"02:16.025 ","End":"02:17.490","Text":"Let\u0027s see about this."},{"Start":"02:17.490 ","End":"02:21.830","Text":"This is the same as 2n squared bigger or equal to n squared plus 2n plus 1."},{"Start":"02:21.830 ","End":"02:27.410","Text":"This is the same as n squared minus 2n is bigger or equal to 1."},{"Start":"02:27.410 ","End":"02:29.210","Text":"Now, we can complete the square,"},{"Start":"02:29.210 ","End":"02:31.685","Text":"n squared minus 2n plus 1 bigger or equal to 2."},{"Start":"02:31.685 ","End":"02:35.045","Text":"Then this is n minus 1 squared bigger or equal to 2."},{"Start":"02:35.045 ","End":"02:38.240","Text":"This is true for n bigger or equal to 3."},{"Start":"02:38.240 ","End":"02:40.325","Text":"If you plug in n equals 0, 1, 2."},{"Start":"02:40.325 ","End":"02:42.260","Text":"From 3 onwards, this is true,"},{"Start":"02:42.260 ","End":"02:44.995","Text":"3 minus 1 squared is 4,"},{"Start":"02:44.995 ","End":"02:46.350","Text":"4 minus 1 squared,"},{"Start":"02:46.350 ","End":"02:48.870","Text":"5 minus 1 squared, they\u0027re all bigger or equal to 2."},{"Start":"02:48.870 ","End":"02:53.580","Text":"Certainly, if n is bigger or equal to 4, then it\u0027s true."},{"Start":"02:53.580 ","End":"02:55.640","Text":"We know that n is bigger or equal to 4,"},{"Start":"02:55.640 ","End":"02:58.165","Text":"that\u0027s part of our assumption."},{"Start":"02:58.165 ","End":"03:00.500","Text":"I forgot to emphasize earlier."},{"Start":"03:00.500 ","End":"03:03.815","Text":"Recall that n is bigger or equal to 4."},{"Start":"03:03.815 ","End":"03:07.310","Text":"This is what we had to prove and we\u0027ve proved it,"},{"Start":"03:07.310 ","End":"03:10.200","Text":"and that concludes this exercise."}],"ID":26626},{"Watched":false,"Name":"Exercise 4","Duration":"4m 38s","ChapterTopicVideoID":25823,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.390","Text":"This exercise is about inequality called Bernoulli\u0027s inequality."},{"Start":"00:06.390 ","End":"00:12.300","Text":"In part a, we have to prove by induction that 1 plus x ^n is bigger or equal to"},{"Start":"00:12.300 ","End":"00:14.190","Text":"1 plus nx for"},{"Start":"00:14.190 ","End":"00:19.395","Text":"all natural n and there\u0027s a condition on x that it\u0027s bigger or equal to minus 1."},{"Start":"00:19.395 ","End":"00:24.090","Text":"In part b, we\u0027ll use part a to prove that 1 plus 1"},{"Start":"00:24.090 ","End":"00:29.250","Text":"over n^n is less than or equal to 1 plus 1 over n plus 1^n plus 1,"},{"Start":"00:29.250 ","End":"00:32.685","Text":"for all natural n. In part a,"},{"Start":"00:32.685 ","End":"00:34.770","Text":"we\u0027ll denote P of n,"},{"Start":"00:34.770 ","End":"00:40.070","Text":"the claim that 1 plus x^n is bigger or equal to 1 plus nx and our task"},{"Start":"00:40.070 ","End":"00:45.440","Text":"is to prove that P of n is true for all n. By induction,"},{"Start":"00:45.440 ","End":"00:47.960","Text":"we\u0027ll start with the base case P of 1."},{"Start":"00:47.960 ","End":"00:51.380","Text":"P of 1 says that 1 plus x ^1 bigger or equal to"},{"Start":"00:51.380 ","End":"00:57.600","Text":"1 plus x and that\u0027s clearly true because they\u0027re the same left and right."},{"Start":"00:57.600 ","End":"00:59.450","Text":"Now in part 2,"},{"Start":"00:59.450 ","End":"01:01.775","Text":"we assume that for some n,"},{"Start":"01:01.775 ","End":"01:04.925","Text":"P of n is true, that\u0027s the induction hypothesis."},{"Start":"01:04.925 ","End":"01:09.440","Text":"In other words, 1 plus x^n bigger or equal to 1 plus nx for this particular n,"},{"Start":"01:09.440 ","End":"01:15.100","Text":"and what we have to do is prove that P of n plus 1 is true for this same n,"},{"Start":"01:15.100 ","End":"01:17.480","Text":"either 1 plus x^n plus 1,"},{"Start":"01:17.480 ","End":"01:20.750","Text":"bigger or equal to 1 plus n plus 1x."},{"Start":"01:20.750 ","End":"01:24.380","Text":"Let\u0027s start with this and see if we can work backwards"},{"Start":"01:24.380 ","End":"01:28.135","Text":"starting from a true statement to derive this."},{"Start":"01:28.135 ","End":"01:32.255","Text":"We can expand the exponent like so."},{"Start":"01:32.255 ","End":"01:34.550","Text":"Now we\u0027ll use the induction hypothesis."},{"Start":"01:34.550 ","End":"01:37.130","Text":"If this is bigger or equal to this,"},{"Start":"01:37.130 ","End":"01:41.870","Text":"then if we succeed in proving that this inequality is true,"},{"Start":"01:41.870 ","End":"01:43.610","Text":"then this inequality is true."},{"Start":"01:43.610 ","End":"01:47.725","Text":"But it\u0027s important that this is bigger or equal to 0."},{"Start":"01:47.725 ","End":"01:52.595","Text":"Let\u0027s expand here, we get this simplify a bit."},{"Start":"01:52.595 ","End":"01:55.745","Text":"This says nx squared bigger or equal to 0,"},{"Start":"01:55.745 ","End":"01:59.845","Text":"and that\u0027s obviously true and this concludes part a."},{"Start":"01:59.845 ","End":"02:04.610","Text":"By the way, there is a variant of Bernoulli\u0027s inequality,"},{"Start":"02:04.610 ","End":"02:08.960","Text":"which can be sharpened to a strict inequality if"},{"Start":"02:08.960 ","End":"02:13.250","Text":"x is bigger than 0 and n is bigger or equal to 2 won\u0027t go through all the steps."},{"Start":"02:13.250 ","End":"02:18.000","Text":"But if you start with bigger than and you assume these 2 facts,"},{"Start":"02:18.000 ","End":"02:20.790","Text":"then it will stay bigger than."},{"Start":"02:20.790 ","End":"02:23.704","Text":"Now on to part b."},{"Start":"02:23.704 ","End":"02:28.410","Text":"We have to prove the following for all n,"},{"Start":"02:28.410 ","End":"02:34.490","Text":"divide both sides by 1 plus 1 over n^n plus"},{"Start":"02:34.490 ","End":"02:40.775","Text":"1 and here we have a minus 1 exponent and here we just put it on the denominator."},{"Start":"02:40.775 ","End":"02:43.490","Text":"I want to do some simplification."},{"Start":"02:43.490 ","End":"02:47.390","Text":"On the left side, we just have n plus 1 over n inside,"},{"Start":"02:47.390 ","End":"02:52.385","Text":"and here we have n plus 2 over n plus 1 in the brackets,"},{"Start":"02:52.385 ","End":"02:55.144","Text":"and n plus 1 over n in the brackets."},{"Start":"02:55.144 ","End":"02:57.650","Text":"This we can simplify,"},{"Start":"02:57.650 ","End":"03:01.660","Text":"divide these 2 fractions and then to the power of n plus 1."},{"Start":"03:01.660 ","End":"03:04.849","Text":"The denominator of the denominator comes into the numerator,"},{"Start":"03:04.849 ","End":"03:06.655","Text":"we end up with this."},{"Start":"03:06.655 ","End":"03:08.929","Text":"The minus 1 means reciprocal,"},{"Start":"03:08.929 ","End":"03:13.280","Text":"so we just flip it and the right-hand side just expand n squared plus 2,"},{"Start":"03:13.280 ","End":"03:16.175","Text":"n, and here n square plus 2n plus 1."},{"Start":"03:16.175 ","End":"03:19.850","Text":"This we can write as n plus 1 minus 1 over n plus"},{"Start":"03:19.850 ","End":"03:24.425","Text":"1 and here also we can do plus 1 minus 1."},{"Start":"03:24.425 ","End":"03:27.485","Text":"You\u0027ll see in a moment where I\u0027m going with this."},{"Start":"03:27.485 ","End":"03:32.180","Text":"This we can write as 1 minus 1 over n plus 1 and here,"},{"Start":"03:32.180 ","End":"03:37.000","Text":"1 minus 1 over n plus 1 squared to the power of."},{"Start":"03:37.000 ","End":"03:43.450","Text":"Let x be this part here so that we have 1 plus x here and"},{"Start":"03:43.450 ","End":"03:46.385","Text":"note that this is bigger or equal to minus 1"},{"Start":"03:46.385 ","End":"03:49.580","Text":"because 1 over this is less than or equal to 1."},{"Start":"03:49.580 ","End":"03:52.855","Text":"We\u0027re going to use Bernoulli\u0027s inequality from part a,"},{"Start":"03:52.855 ","End":"03:58.900","Text":"1 plus n plus 1x is less than or equal to 1 plus x^n plus 1."},{"Start":"03:58.900 ","End":"04:06.365","Text":"We can rewrite this as 1 plus n plus 1x because if you multiply this by n plus 1,"},{"Start":"04:06.365 ","End":"04:08.300","Text":"you get minus 1 over n plus 1,"},{"Start":"04:08.300 ","End":"04:09.575","Text":"which is what we have here."},{"Start":"04:09.575 ","End":"04:11.165","Text":"On the left-hand side,"},{"Start":"04:11.165 ","End":"04:13.370","Text":"we have 1 plus x,"},{"Start":"04:13.370 ","End":"04:16.970","Text":"because x is minus this to the power of n plus 1."},{"Start":"04:16.970 ","End":"04:21.320","Text":"Now this follows from Bernoulli\u0027s inequality with part a."},{"Start":"04:21.320 ","End":"04:25.085","Text":"If you look what it says, it\u0027s the same thing just with n instead of n plus 1."},{"Start":"04:25.085 ","End":"04:28.025","Text":"But doesn\u0027t matter, we can replace n by n plus 1."},{"Start":"04:28.025 ","End":"04:30.205","Text":"This is true, therefore, this is true."},{"Start":"04:30.205 ","End":"04:32.040","Text":"It all works backwards."},{"Start":"04:32.040 ","End":"04:39.210","Text":"We prove this, and that\u0027s the induction part that completes part b and we\u0027re done."}],"ID":26627},{"Watched":false,"Name":"Exercise 5","Duration":"2m 59s","ChapterTopicVideoID":25824,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.740","Text":"In this exercise, we have to prove"},{"Start":"00:02.740 ","End":"00:07.025","Text":"an inequality that\u0027s very similar to Bernoulli\u0027s inequality."},{"Start":"00:07.025 ","End":"00:09.705","Text":"That is for all natural numbers, n,"},{"Start":"00:09.705 ","End":"00:14.035","Text":"1 minus x to the n is bigger than 1 over 1 plus nx,"},{"Start":"00:14.035 ","End":"00:18.310","Text":"provided that x is between 0 and 1."},{"Start":"00:18.310 ","End":"00:20.820","Text":"We do it by induction,"},{"Start":"00:20.820 ","End":"00:27.030","Text":"we let p of n be the statement 1 minus x to the n bigger than 1 over 1 plus nx."},{"Start":"00:27.030 ","End":"00:29.970","Text":"We have to prove that p of n is true for all n. We\u0027ll"},{"Start":"00:29.970 ","End":"00:33.340","Text":"start with the base case p of 1. Check that that\u0027s true."},{"Start":"00:33.340 ","End":"00:35.350","Text":"If you plug in n equals 1,"},{"Start":"00:35.350 ","End":"00:41.235","Text":"then we get 1 minus x to the 1 is less than 1 over 1 plus x."},{"Start":"00:41.235 ","End":"00:42.885","Text":"Is this true?"},{"Start":"00:42.885 ","End":"00:44.580","Text":"I say it is,"},{"Start":"00:44.580 ","End":"00:48.360","Text":"because it follows from 1 minus x,"},{"Start":"00:48.360 ","End":"00:50.980","Text":"1 plus x is less than 1."},{"Start":"00:50.980 ","End":"00:53.600","Text":"Note that 1 plus x is positive,"},{"Start":"00:53.600 ","End":"00:55.160","Text":"so this is equivalent to this."},{"Start":"00:55.160 ","End":"00:57.155","Text":"Now why is this true?"},{"Start":"00:57.155 ","End":"01:01.585","Text":"This, if you expand it is 1 minus x squared,"},{"Start":"01:01.585 ","End":"01:04.040","Text":"and 1 minus x squared is less than 1,"},{"Start":"01:04.040 ","End":"01:05.690","Text":"because x is positive."},{"Start":"01:05.690 ","End":"01:07.400","Text":"So this is true, so this true,"},{"Start":"01:07.400 ","End":"01:10.280","Text":"so this true, so p of 1 is true."},{"Start":"01:10.280 ","End":"01:13.580","Text":"Now, the induction hypothesis,"},{"Start":"01:13.580 ","End":"01:17.210","Text":"we\u0027ll assume that p of n is true for some"},{"Start":"01:17.210 ","End":"01:23.140","Text":"n. That means that for this n, this inequality holds."},{"Start":"01:23.140 ","End":"01:27.200","Text":"We have to prove that p of n plus 1 is true for the"},{"Start":"01:27.200 ","End":"01:30.980","Text":"same n. What we have to do, in other words,"},{"Start":"01:30.980 ","End":"01:37.040","Text":"is to prove that 1 minus x to the n plus 1 is bigger than 1 over 1 plus n plus 1 x,"},{"Start":"01:37.040 ","End":"01:43.160","Text":"we just put n plus 1 instead of n. This is what we have to show,"},{"Start":"01:43.160 ","End":"01:46.085","Text":"and we can use this to show it."},{"Start":"01:46.085 ","End":"01:50.930","Text":"Expand this using the rule of exponents."},{"Start":"01:50.930 ","End":"01:55.625","Text":"Well, this is identical to this but implies it."},{"Start":"01:55.625 ","End":"02:02.375","Text":"Now this by the induction hypothesis is bigger than this."},{"Start":"02:02.375 ","End":"02:05.405","Text":"If we can prove this inequality,"},{"Start":"02:05.405 ","End":"02:11.405","Text":"then this inequality will follow also because 1 minus x is positive."},{"Start":"02:11.405 ","End":"02:14.870","Text":"We can multiply an inequality by a positive number."},{"Start":"02:14.870 ","End":"02:17.104","Text":"This is what we have to show."},{"Start":"02:17.104 ","End":"02:20.690","Text":"Now, both these denominators are positive,"},{"Start":"02:20.690 ","End":"02:24.350","Text":"so we can multiply both sides by the product of these denominators,"},{"Start":"02:24.350 ","End":"02:28.340","Text":"like cross-multiplying, this will come here and this will go here,"},{"Start":"02:28.340 ","End":"02:30.410","Text":"this is what we get."},{"Start":"02:30.410 ","End":"02:35.285","Text":"Open the brackets and simplify a bit,"},{"Start":"02:35.285 ","End":"02:38.385","Text":"simplify some more take the x squared out."},{"Start":"02:38.385 ","End":"02:44.210","Text":"At this point it\u0027s clear that this is positive because x is positive and n is positive."},{"Start":"02:44.210 ","End":"02:45.650","Text":"So n plus 1 is positive."},{"Start":"02:45.650 ","End":"02:46.970","Text":"Because this is positive, this is true,"},{"Start":"02:46.970 ","End":"02:48.260","Text":"this is true, this is true, this is true,"},{"Start":"02:48.260 ","End":"02:53.190","Text":"this is true and eventually we get to p of n plus 1 is true."},{"Start":"02:53.190 ","End":"02:55.470","Text":"That\u0027s the induction step,"},{"Start":"02:55.470 ","End":"02:59.230","Text":"and that concludes this exercise."}],"ID":26628},{"Watched":false,"Name":"Exercise 6","Duration":"3m 45s","ChapterTopicVideoID":25825,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this exercise, we\u0027re going to prove by induction that n"},{"Start":"00:03.540 ","End":"00:06.510","Text":"factorial is less than or equal to n plus 1"},{"Start":"00:06.510 ","End":"00:10.350","Text":"over 2,^n for all natural numbers n."},{"Start":"00:10.350 ","End":"00:14.055","Text":"This n with an exclamation mark is n factorial,"},{"Start":"00:14.055 ","End":"00:16.125","Text":"I\u0027m hoping you remember it."},{"Start":"00:16.125 ","End":"00:23.585","Text":"In case not n factorial is n times n minus 1 and so on times 3 times 2 times 1."},{"Start":"00:23.585 ","End":"00:26.590","Text":"For example, 7 factorial is this."},{"Start":"00:26.590 ","End":"00:30.905","Text":"As a useful rule that we\u0027ll use,"},{"Start":"00:30.905 ","End":"00:34.985","Text":"n factorial is n times n minus 1 factorial."},{"Start":"00:34.985 ","End":"00:36.910","Text":"For example, 7 factorial,"},{"Start":"00:36.910 ","End":"00:41.420","Text":"you can look at it as 7 and the rest of it is 6 factorial."},{"Start":"00:41.420 ","End":"00:44.070","Text":"That\u0027s an example of this rule."},{"Start":"00:45.110 ","End":"00:48.500","Text":"P of n will denote the statement"},{"Start":"00:48.500 ","End":"00:51.500","Text":"for a particular end that this inequality holds."},{"Start":"00:51.500 ","End":"00:55.070","Text":"We want to show that P of n is true for all n. We"},{"Start":"00:55.070 ","End":"00:59.140","Text":"start by checking the base case that P of 1 is true."},{"Start":"00:59.140 ","End":"01:03.710","Text":"That says that 1 factorial, well,"},{"Start":"01:03.710 ","End":"01:05.630","Text":"it says what is written here,"},{"Start":"01:05.630 ","End":"01:09.800","Text":"and that\u0027s true because both sides evaluate to 1."},{"Start":"01:09.800 ","End":"01:16.535","Text":"Now, the induction hypothesis that P of n is true for a particular n,"},{"Start":"01:16.535 ","End":"01:18.710","Text":"and we assume this."},{"Start":"01:18.710 ","End":"01:24.440","Text":"n factorial is less than n plus 1 over 2,^n for this n. Now we have to"},{"Start":"01:24.440 ","End":"01:31.910","Text":"prove from this that P of n plus 1 is true for the same n. In other words,"},{"Start":"01:31.910 ","End":"01:35.630","Text":"we have to prove that n plus 1 factorial is less than or equal"},{"Start":"01:35.630 ","End":"01:40.135","Text":"to n plus 1 plus 1 over 2,^n plus 1."},{"Start":"01:40.135 ","End":"01:43.880","Text":"We\u0027re allowed to use this to prove this."},{"Start":"01:43.880 ","End":"01:47.450","Text":"Let\u0027s see, let\u0027s work backwards."},{"Start":"01:47.450 ","End":"01:52.610","Text":"n plus 1 factorial less than or equal to, what it says here."},{"Start":"01:52.610 ","End":"01:58.759","Text":"Now, break this up according to what I just said above."},{"Start":"01:58.759 ","End":"02:02.975","Text":"This is equal to n plus 1 times n factorial."},{"Start":"02:02.975 ","End":"02:05.495","Text":"Right-hand side is untouched."},{"Start":"02:05.495 ","End":"02:08.315","Text":"If the induction hypothesis is true,"},{"Start":"02:08.315 ","End":"02:13.925","Text":"then n factorial is less than or equal to n plus 1 over 2,^n."},{"Start":"02:13.925 ","End":"02:16.930","Text":"If I prove this inequality,"},{"Start":"02:16.930 ","End":"02:22.505","Text":"I replace the left-hand side by something smaller possibly then it will also be true."},{"Start":"02:22.505 ","End":"02:25.340","Text":"If we can prove this, then this will be true."},{"Start":"02:25.340 ","End":"02:29.875","Text":"Now here we can spend on the denominator, 2^n,"},{"Start":"02:29.875 ","End":"02:32.950","Text":"and here n plus 1,^n times n plus 1,"},{"Start":"02:32.950 ","End":"02:35.675","Text":"so it\u0027s n plus 1,^ n plus 1."},{"Start":"02:35.675 ","End":"02:41.975","Text":"On the right-hand side, we\u0027ll just decompose the denominator as 2^n times 2."},{"Start":"02:41.975 ","End":"02:45.110","Text":"Now the 2^n cancels with 2^n."},{"Start":"02:45.110 ","End":"02:48.995","Text":"The 2 here can be brought over to the left-hand side,"},{"Start":"02:48.995 ","End":"02:51.860","Text":"and this n plus 1,^n plus 1 can be brought into"},{"Start":"02:51.860 ","End":"02:56.105","Text":"the denominator and then we\u0027ll collect it under the exponent n plus 1."},{"Start":"02:56.105 ","End":"02:59.570","Text":"We end up with this and if this is true,"},{"Start":"02:59.570 ","End":"03:06.780","Text":"this is true and then this can be written as 1 plus 1 over n plus 1, what\u0027s inside here."},{"Start":"03:06.880 ","End":"03:10.970","Text":"Now we can use Bernoulli\u0027s inequality."},{"Start":"03:10.970 ","End":"03:14.134","Text":"This is what Bernoulli\u0027s inequality says."},{"Start":"03:14.134 ","End":"03:18.350","Text":"We had it with n, I\u0027ll do it with m because we\u0027ve used up n already."},{"Start":"03:18.350 ","End":"03:23.050","Text":"This is true for natural m and for x bigger or equal to minus 1."},{"Start":"03:23.050 ","End":"03:26.270","Text":"We\u0027ll let x be 1 over n plus 1,"},{"Start":"03:26.270 ","End":"03:29.045","Text":"and we let m be n plus 1."},{"Start":"03:29.045 ","End":"03:34.565","Text":"Then 1 plus mx is 1 plus n plus 1 over n plus 1, which is 2."},{"Start":"03:34.565 ","End":"03:40.510","Text":"This is just 1 plus x,^m."},{"Start":"03:40.510 ","End":"03:45.760","Text":"That\u0027s what we wanted and that means that we\u0027re done."}],"ID":26629},{"Watched":false,"Name":"Exercise 7","Duration":"3m 3s","ChapterTopicVideoID":25826,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.950","Text":"In this exercise,"},{"Start":"00:01.950 ","End":"00:05.010","Text":"we define a sequence a _ n recursively."},{"Start":"00:05.010 ","End":"00:08.730","Text":"We let a_1 be the square root of 2,"},{"Start":"00:08.730 ","End":"00:13.320","Text":"and we define a_n plus 1 in terms of a_n as follows."},{"Start":"00:13.320 ","End":"00:16.575","Text":"a_n plus 1 equals the square root of a_n plus 2."},{"Start":"00:16.575 ","End":"00:19.200","Text":"We have to prove by induction 2 things."},{"Start":"00:19.200 ","End":"00:27.105","Text":"First of all, that the sequence a_n is bounded above by 2, and secondly,"},{"Start":"00:27.105 ","End":"00:34.350","Text":"that the sequence is increasing a_n is less than or equal to a_n plus 1 for all n. Let\u0027s"},{"Start":"00:34.350 ","End":"00:41.895","Text":"start with Part a and let P of n be the property that a_n is less than or equal to 2."},{"Start":"00:41.895 ","End":"00:45.480","Text":"We want to show that P of n is true for all n by induction,"},{"Start":"00:45.480 ","End":"00:48.270","Text":"so we start with the base case P of 1."},{"Start":"00:48.270 ","End":"00:52.365","Text":"What this says is that a_1 is less than or equal to 2."},{"Start":"00:52.365 ","End":"00:55.260","Text":"This is true because a_1 is square root of 2,"},{"Start":"00:55.260 ","End":"00:57.750","Text":"which is definitely less than or equal to 2."},{"Start":"00:57.750 ","End":"00:59.910","Text":"Now the induction hypothesis,"},{"Start":"00:59.910 ","End":"01:04.005","Text":"so we assume p_n is true for a particular n. From this,"},{"Start":"01:04.005 ","End":"01:07.740","Text":"we want to prove that P of n plus 1 is true."},{"Start":"01:07.740 ","End":"01:12.390","Text":"This says that a_n plus 1 is less than or equal to 2."},{"Start":"01:12.390 ","End":"01:16.335","Text":"We want to use this to prove this."},{"Start":"01:16.335 ","End":"01:21.495","Text":"a_n plus 1 by the recursive definition is the square root of a_n plus 2."},{"Start":"01:21.495 ","End":"01:25.140","Text":"This is less than or equal to 2 by the induction hypothesis,"},{"Start":"01:25.140 ","End":"01:27.570","Text":"so a_n plus 2 is less than or equal to 2 plus 2."},{"Start":"01:27.570 ","End":"01:30.690","Text":"When you take the square root, it\u0027s still inequality."},{"Start":"01:30.690 ","End":"01:33.270","Text":"This is square root of 4, which is 2,"},{"Start":"01:33.270 ","End":"01:37.395","Text":"so a_n plus 1 is less than or equal to 2."},{"Start":"01:37.395 ","End":"01:39.975","Text":"That completes Part a."},{"Start":"01:39.975 ","End":"01:44.850","Text":"Part b, we want to show that a_n is increasing as was each a_n is"},{"Start":"01:44.850 ","End":"01:49.740","Text":"less than or equal to the subsequent a_n again will show it by induction."},{"Start":"01:49.740 ","End":"01:54.870","Text":"P of 1 says that a_1 is less than or equal to a_2. Is that true?"},{"Start":"01:54.870 ","End":"01:56.580","Text":"Well, a_1 is square root of 2,"},{"Start":"01:56.580 ","End":"01:58.245","Text":"and a_2, if you compute it,"},{"Start":"01:58.245 ","End":"02:02.655","Text":"comes out to be the square root of the square root of 2 plus 2."},{"Start":"02:02.655 ","End":"02:05.670","Text":"Then a_1 is less than or equal to a_2 because what\u0027s under"},{"Start":"02:05.670 ","End":"02:08.310","Text":"the square root sign is less than or equal to 2,"},{"Start":"02:08.310 ","End":"02:11.755","Text":"is less than or equal to 2 plus the square root of 2."},{"Start":"02:11.755 ","End":"02:15.950","Text":"Now, assume that P of n is true for some particular"},{"Start":"02:15.950 ","End":"02:19.780","Text":"n. In other words that a_n is less than or equal to a_n plus 1."},{"Start":"02:19.780 ","End":"02:24.900","Text":"From this we\u0027ll prove that P of n plus 1 is true for this n. In other words,"},{"Start":"02:24.900 ","End":"02:28.950","Text":"that a_n plus 1 is less than or equal to a_n plus 1 plus 1,"},{"Start":"02:28.950 ","End":"02:30.630","Text":"which is a_n plus 2,"},{"Start":"02:30.630 ","End":"02:33.630","Text":"so we use this to prove this."},{"Start":"02:33.630 ","End":"02:38.545","Text":"a_n plus 2 is a square root of a_n plus 1 plus 2."},{"Start":"02:38.545 ","End":"02:40.520","Text":"By the induction hypothesis,"},{"Start":"02:40.520 ","End":"02:42.515","Text":"this is bigger or equal to this,"},{"Start":"02:42.515 ","End":"02:44.690","Text":"so even if you add 2 and take the square root,"},{"Start":"02:44.690 ","End":"02:46.550","Text":"it\u0027s still an inequality."},{"Start":"02:46.550 ","End":"02:48.670","Text":"This is equal to a_n plus 1,"},{"Start":"02:48.670 ","End":"02:52.995","Text":"so altogether we have a_n plus 2 bigger or equal to a_n plus 1."},{"Start":"02:52.995 ","End":"02:56.355","Text":"Or if you\u0027d like a_n plus 1 is less than or equal to a_n plus 2."},{"Start":"02:56.355 ","End":"02:59.130","Text":"That\u0027s the induction part here,"},{"Start":"02:59.130 ","End":"03:03.490","Text":"so that completes Part b and we\u0027re done."}],"ID":26630},{"Watched":false,"Name":"Exercise 8","Duration":"5m 26s","ChapterTopicVideoID":25827,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.105","Text":"In this exercise, we define a sequence a_n recursively as follows."},{"Start":"00:06.105 ","End":"00:09.315","Text":"We\u0027re given a_1 and a_2,"},{"Start":"00:09.315 ","End":"00:12.690","Text":"and we generate subsequent members"},{"Start":"00:12.690 ","End":"00:15.645","Text":"from the two previous members,"},{"Start":"00:15.645 ","End":"00:20.445","Text":"a_n and a_ n plus 1 gave us a_n plus 2 using this formula."},{"Start":"00:20.445 ","End":"00:22.290","Text":"Three parts. First of all,"},{"Start":"00:22.290 ","End":"00:25.800","Text":"compute a_3, a_4, a_5,"},{"Start":"00:25.800 ","End":"00:28.440","Text":"then to prove by induction,"},{"Start":"00:28.440 ","End":"00:31.490","Text":"that\u0027s going to be a modified form of induction,"},{"Start":"00:31.490 ","End":"00:34.340","Text":"that a_n equals n_2 minus 2n."},{"Start":"00:34.340 ","End":"00:36.860","Text":"Then we\u0027ll verify this on the a_3,"},{"Start":"00:36.860 ","End":"00:39.335","Text":"a_4, a_5 that we computed."},{"Start":"00:39.335 ","End":"00:43.459","Text":"A proof by induction is very suited to recursive sequences."},{"Start":"00:43.459 ","End":"00:44.630","Text":"It occurs a lot."},{"Start":"00:44.630 ","End":"00:46.255","Text":"I was just mentioning that."},{"Start":"00:46.255 ","End":"00:47.910","Text":"Let\u0027s start with part a."},{"Start":"00:47.910 ","End":"00:50.715","Text":"Let\u0027s compute a_3."},{"Start":"00:50.715 ","End":"00:55.380","Text":"By the formula, we let n equals 1,"},{"Start":"00:55.380 ","End":"00:57.450","Text":"and then we get a_3 here,"},{"Start":"00:57.450 ","End":"01:00.405","Text":"and a_2 here and a_1 here."},{"Start":"01:00.405 ","End":"01:04.560","Text":"This is what we get, then substitute a_2 is 0,"},{"Start":"01:04.560 ","End":"01:06.570","Text":"and a_1 is minus 1,"},{"Start":"01:06.570 ","End":"01:08.580","Text":"so it comes out to be 3."},{"Start":"01:08.580 ","End":"01:15.990","Text":"Then we can let n equals 2 and compute a_4 from a_3 and a_2 comes out 8."},{"Start":"01:15.990 ","End":"01:19.560","Text":"Similarly, a_5 is computed from a_4 and a_3,"},{"Start":"01:19.560 ","End":"01:23.055","Text":"and a_4, we have here is 8, a_3 is 3."},{"Start":"01:23.055 ","End":"01:25.410","Text":"Here\u0027s the 8, and here\u0027s the 3."},{"Start":"01:25.410 ","End":"01:27.435","Text":"Anyway, we get 15."},{"Start":"01:27.435 ","End":"01:34.585","Text":"On to part b, where we have to prove the proposition that a_n equals n_2 minus 2n."},{"Start":"01:34.585 ","End":"01:37.420","Text":"We have to prove that P of n, which is this,"},{"Start":"01:37.420 ","End":"01:40.030","Text":"is true for all n. Now we\u0027re going to modify"},{"Start":"01:40.030 ","End":"01:44.020","Text":"the induction instead of starting with just P of 1,"},{"Start":"01:44.020 ","End":"01:49.660","Text":"the base case will be that P of 1 is true and P of 2 is true."},{"Start":"01:49.660 ","End":"01:54.485","Text":"We\u0027re modifying this because each element depends on two previous ones."},{"Start":"01:54.485 ","End":"01:57.025","Text":"Then this is just the plan."},{"Start":"01:57.025 ","End":"02:02.680","Text":"We\u0027re going to assume that P of n and P of n plus 1 are both true for a particular"},{"Start":"02:02.680 ","End":"02:10.050","Text":"n. We\u0027ll use this to prove that P of n plus 2 is true for this n. That\u0027s the plan."},{"Start":"02:10.050 ","End":"02:13.250","Text":"What\u0027s the logic behind this modified induction?"},{"Start":"02:13.250 ","End":"02:18.520","Text":"Let\u0027s see. P of 1 and P of 2, we check directly."},{"Start":"02:18.520 ","End":"02:22.955","Text":"Because P of 1 is true and P of 2 is true,"},{"Start":"02:22.955 ","End":"02:24.860","Text":"then P of 3 is true."},{"Start":"02:24.860 ","End":"02:27.575","Text":"From n and n plus 1, we get n plus 2."},{"Start":"02:27.575 ","End":"02:29.485","Text":"If n is 1,"},{"Start":"02:29.485 ","End":"02:32.250","Text":"then 1 and 2 gives us 3."},{"Start":"02:32.250 ","End":"02:34.155","Text":"Once 3 is true,"},{"Start":"02:34.155 ","End":"02:36.150","Text":"then we know that P of 2 is true,"},{"Start":"02:36.150 ","End":"02:37.710","Text":"and P of 3 is true."},{"Start":"02:37.710 ","End":"02:41.630","Text":"From these, we can get P of 4 by letting n equals 2."},{"Start":"02:41.630 ","End":"02:43.490","Text":"Here we have 2, 3, 4,"},{"Start":"02:43.490 ","End":"02:47.370","Text":"and then 3 and 4 gave us 5, and so on."},{"Start":"02:47.370 ","End":"02:50.420","Text":"Eventually, P of everything will be true,"},{"Start":"02:50.420 ","End":"02:55.305","Text":"P of n, so that\u0027s why this modified induction works."},{"Start":"02:55.305 ","End":"02:57.990","Text":"Let\u0027s actually implement the three steps."},{"Start":"02:57.990 ","End":"03:01.025","Text":"First of all, P of 1, what it says,"},{"Start":"03:01.025 ","End":"03:04.230","Text":"here we have it and then n equals 1,"},{"Start":"03:04.230 ","End":"03:08.445","Text":"then we want a_1 equals 1^2 minus 2 times 1."},{"Start":"03:08.445 ","End":"03:10.640","Text":"Both sides come out minus 1."},{"Start":"03:10.640 ","End":"03:13.030","Text":"Yes, this is true."},{"Start":"03:13.030 ","End":"03:19.905","Text":"P of 2, it says that 2^2 minus twice 2 is a_2."},{"Start":"03:19.905 ","End":"03:21.375","Text":"A_2 is 0."},{"Start":"03:21.375 ","End":"03:22.740","Text":"2^2 minus twice 2 is 0,"},{"Start":"03:22.740 ","End":"03:24.855","Text":"so these are equal also."},{"Start":"03:24.855 ","End":"03:27.705","Text":"That\u0027s Step 1. Step 2,"},{"Start":"03:27.705 ","End":"03:29.400","Text":"we\u0027re assuming two things."},{"Start":"03:29.400 ","End":"03:31.965","Text":"Done it for n and n plus 1."},{"Start":"03:31.965 ","End":"03:39.950","Text":"We have a_n is n_2 minus 2_n and a_n plus 1 is n plus 1^2 minus twice n plus 1."},{"Start":"03:39.950 ","End":"03:41.090","Text":"What do we have to prove?"},{"Start":"03:41.090 ","End":"03:43.340","Text":"We have to prove P of n plus 2,"},{"Start":"03:43.340 ","End":"03:48.380","Text":"which says that a_n plus 2 is n plus 2^2 minus twice n plus 2."},{"Start":"03:48.380 ","End":"03:50.120","Text":"From this and this,"},{"Start":"03:50.120 ","End":"03:52.060","Text":"you want to derive this."},{"Start":"03:52.060 ","End":"03:59.240","Text":"We have the recursion formula that a_n plus 2 is based on a_n plus 1 and a_n as follows."},{"Start":"03:59.240 ","End":"04:04.720","Text":"Now we can substitute a_n plus 1 from here, a_n from here."},{"Start":"04:04.720 ","End":"04:06.765","Text":"It\u0027s just a bit of algebra."},{"Start":"04:06.765 ","End":"04:09.240","Text":"I\u0027ll just let you take a look,"},{"Start":"04:09.240 ","End":"04:12.690","Text":"and we get to n_2 plus 2n."},{"Start":"04:12.690 ","End":"04:14.295","Text":"On the other hand,"},{"Start":"04:14.295 ","End":"04:17.044","Text":"now we\u0027ll work on the right-hand side."},{"Start":"04:17.044 ","End":"04:21.320","Text":"We\u0027ve already checked that the left-hand side is n_2 plus 2n,"},{"Start":"04:21.320 ","End":"04:22.610","Text":"but the right-hand side,"},{"Start":"04:22.610 ","End":"04:23.960","Text":"if you expand it,"},{"Start":"04:23.960 ","End":"04:27.605","Text":"you also get n^2 plus 2n because you get n_2 from here,"},{"Start":"04:27.605 ","End":"04:31.545","Text":"we have plus 4n and minus 2n."},{"Start":"04:31.545 ","End":"04:35.165","Text":"Also, the 2^2 is 4 and 2 times 2 is 4 cancel."},{"Start":"04:35.165 ","End":"04:37.990","Text":"This is what we get, and these 2 are the same."},{"Start":"04:37.990 ","End":"04:40.545","Text":"That concludes part b of the exercise."},{"Start":"04:40.545 ","End":"04:45.049","Text":"Part C, we have to verify for a, 3, 4, and 5."},{"Start":"04:45.049 ","End":"04:46.415","Text":"Let\u0027s see."},{"Start":"04:46.415 ","End":"04:48.485","Text":"If we plug in."},{"Start":"04:48.485 ","End":"04:51.260","Text":"This is the formula that we\u0027re using."},{"Start":"04:51.260 ","End":"04:55.060","Text":"3^2 minus twice 3 is 3."},{"Start":"04:55.060 ","End":"04:56.985","Text":"If you go back and check,"},{"Start":"04:56.985 ","End":"04:59.270","Text":"this is what a_3 is."},{"Start":"04:59.270 ","End":"05:02.080","Text":"It\u0027s off screen, but this is what it was."},{"Start":"05:02.080 ","End":"05:06.240","Text":"N equals 4, 4^2 minus twice 4 is 8,"},{"Start":"05:06.240 ","End":"05:07.830","Text":"and that is equal to a_4."},{"Start":"05:07.830 ","End":"05:09.885","Text":"Again, go back and check."},{"Start":"05:09.885 ","End":"05:12.964","Text":"5^2 minus twice 5 is 15,"},{"Start":"05:12.964 ","End":"05:16.460","Text":"and that is equal to a_5. You know what?"},{"Start":"05:16.460 ","End":"05:20.345","Text":"I\u0027ll just scroll back and you can see that we have 3, 8 and 15."},{"Start":"05:20.345 ","End":"05:23.495","Text":"Where are we? 3,"},{"Start":"05:23.495 ","End":"05:27.120","Text":"8, and 15. We\u0027re done."}],"ID":26631},{"Watched":false,"Name":"Exercise 9","Duration":"4m 27s","ChapterTopicVideoID":25828,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.755","Text":"In this exercise, we have a recursive sequence where we\u0027re given a_1 and a_2 explicitly,"},{"Start":"00:07.755 ","End":"00:14.055","Text":"and then we define subsequent elements in terms of the 2 previous ones."},{"Start":"00:14.055 ","End":"00:21.330","Text":"N plus 2 is defined in terms of the previous a_n plus 1 and the 1 before that a_n."},{"Start":"00:21.330 ","End":"00:26.220","Text":"Our task is to prove by induction the following formula for a _n."},{"Start":"00:26.220 ","End":"00:29.100","Text":"It\u0027s not going to be the usual induction,"},{"Start":"00:29.100 ","End":"00:30.900","Text":"we have to modify it a bit."},{"Start":"00:30.900 ","End":"00:35.540","Text":"If the sequence was defined so that each element is given"},{"Start":"00:35.540 ","End":"00:39.290","Text":"in terms of the 1 previous 1 then we use regular induction"},{"Start":"00:39.290 ","End":"00:41.600","Text":"but because it\u0027s defined of 2 previous ones,"},{"Start":"00:41.600 ","End":"00:43.295","Text":"we\u0027re going to modify the induction."},{"Start":"00:43.295 ","End":"00:45.380","Text":"Here\u0027s what we\u0027ll do. First of all,"},{"Start":"00:45.380 ","End":"00:47.780","Text":"let P of n be the claim"},{"Start":"00:47.780 ","End":"00:49.760","Text":"that this is true for a particular n."},{"Start":"00:49.760 ","End":"00:52.580","Text":"The induction will modify as follows;"},{"Start":"00:52.580 ","End":"00:56.840","Text":"is first check that P of 1 is true and P of 2 is true,"},{"Start":"00:56.840 ","End":"00:59.990","Text":"then the induction hypothesis will be to assume that"},{"Start":"00:59.990 ","End":"01:04.765","Text":"both P of n and P of n plus 1 are true,"},{"Start":"01:04.765 ","End":"01:12.980","Text":"and the induction step will be to prove from these that P of n plus 2 is true."},{"Start":"01:12.980 ","End":"01:17.415","Text":"What\u0027s the rationale behind this modified induction? Let\u0027s see."},{"Start":"01:17.415 ","End":"01:24.180","Text":"P of 1 is true and P of 2 is true because we\u0027re checking that directly."},{"Start":"01:24.500 ","End":"01:27.750","Text":"From the truth for n and n plus 1,"},{"Start":"01:27.750 ","End":"01:29.250","Text":"we get n plus 2."},{"Start":"01:29.250 ","End":"01:30.820","Text":"From 1 and 2,"},{"Start":"01:30.820 ","End":"01:35.045","Text":"we get 3, from 2 and 3, we get 4."},{"Start":"01:35.045 ","End":"01:37.955","Text":"From 3 and 4, we get 5."},{"Start":"01:37.955 ","End":"01:39.290","Text":"If we keep going on,"},{"Start":"01:39.290 ","End":"01:41.015","Text":"we\u0027ll reach any n,"},{"Start":"01:41.015 ","End":"01:45.845","Text":"so P of n will be true for every n. That\u0027s the modified induction."},{"Start":"01:45.845 ","End":"01:48.575","Text":"Let\u0027s now actually do these 3 steps."},{"Start":"01:48.575 ","End":"01:53.550","Text":"P of 1 claims that a_1 is 1/6,"},{"Start":"01:53.550 ","End":"01:56.445","Text":"3 to the 1 minus 1/2 minus 1 to the 1."},{"Start":"01:56.445 ","End":"02:03.675","Text":"Both sides come out to be 1 if you calculate it, so that\u0027s okay."},{"Start":"02:03.675 ","End":"02:07.350","Text":"Then P of 2 says a_2 equals 1/6,"},{"Start":"02:07.350 ","End":"02:09.745","Text":"3 squared minus 1/2 minus 1 to the 2."},{"Start":"02:09.745 ","End":"02:12.140","Text":"Again, both sides come out to be 1,"},{"Start":"02:12.140 ","End":"02:15.455","Text":"so they\u0027re equal, so that\u0027s also okay."},{"Start":"02:15.455 ","End":"02:18.285","Text":"Second step, we\u0027re going to assume 2 things."},{"Start":"02:18.285 ","End":"02:20.660","Text":"We\u0027re going to assume that P of n is true,"},{"Start":"02:20.660 ","End":"02:22.610","Text":"ie that this is true,"},{"Start":"02:22.610 ","End":"02:25.880","Text":"and P of n plus 1, which gives us this."},{"Start":"02:25.880 ","End":"02:28.030","Text":"Where we have n plus 1, n plus 1,"},{"Start":"02:28.030 ","End":"02:29.680","Text":"n plus 1 in place of n."},{"Start":"02:29.680 ","End":"02:37.550","Text":"Our task is to prove using these 2 that P of n plus 2 is true, in other words,"},{"Start":"02:37.550 ","End":"02:42.905","Text":"the following where we have n plus 2 in place of n. To rephrase this,"},{"Start":"02:42.905 ","End":"02:45.800","Text":"what it says is that a_n plus 2 is,"},{"Start":"02:45.800 ","End":"02:48.935","Text":"here we can take 3 squared out, which is 9."},{"Start":"02:48.935 ","End":"02:51.775","Text":"9 over 6 is 3 over 2."},{"Start":"02:51.775 ","End":"02:55.430","Text":"Here we can take minus 1 squared times minus 1 to the n."},{"Start":"02:55.430 ","End":"03:01.820","Text":"Minus 1 squared is 1 so you can just knock off the 2, and that\u0027s okay."},{"Start":"03:01.820 ","End":"03:06.495","Text":"This is what we\u0027re going to show given this and this."},{"Start":"03:06.495 ","End":"03:13.210","Text":"Let\u0027s see, for the recursion formula tells us that a_n plus 2 is 2a_n plus 1 plus 3a_n."},{"Start":"03:13.210 ","End":"03:21.360","Text":"We\u0027ll substitute a_n plus 1 here and a_n is this and then do a bit of algebra."},{"Start":"03:21.360 ","End":"03:23.040","Text":"2 over 6 is 1/3,"},{"Start":"03:23.040 ","End":"03:26.880","Text":"so that changes 3 to the n plus 1 down to 3 to the n."},{"Start":"03:26.880 ","End":"03:32.550","Text":"Here we can take a minus 1 out and make it minus 1 to the n,"},{"Start":"03:32.550 ","End":"03:34.340","Text":"and the minus cancels with the minus,"},{"Start":"03:34.340 ","End":"03:35.510","Text":"making it a plus."},{"Start":"03:35.510 ","End":"03:37.835","Text":"Now the second bracket,"},{"Start":"03:37.835 ","End":"03:40.250","Text":"we\u0027ll open it up with the distributive law."},{"Start":"03:40.250 ","End":"03:46.530","Text":"3 times 1/6 is 3/6 is 1/2 3 to the n,"},{"Start":"03:46.530 ","End":"03:49.785","Text":"so that\u0027s this 1/2 3 to the n. Here,"},{"Start":"03:49.785 ","End":"03:56.360","Text":"3 times minus 1/2 is minus 3 over 2 minus 1 to the n. This is the expression we get."},{"Start":"03:56.360 ","End":"04:01.655","Text":"Now let\u0027s collect together the 3 to the n separately and the minus 1 to the n separately."},{"Start":"04:01.655 ","End":"04:05.390","Text":"Here we have 1 times 3 to the n plus 1/2 3 to the n,"},{"Start":"04:05.390 ","End":"04:09.925","Text":"so it\u0027s 1 1/2 3 to the n. 1 1/2 is 3/2."},{"Start":"04:09.925 ","End":"04:11.640","Text":"Here minus 1 to the n,"},{"Start":"04:11.640 ","End":"04:13.395","Text":"we have 1 time,"},{"Start":"04:13.395 ","End":"04:15.750","Text":"minus 1 1/2 times,"},{"Start":"04:15.750 ","End":"04:17.779","Text":"so it\u0027s minus 1/2 times."},{"Start":"04:17.779 ","End":"04:20.810","Text":"If you look here and you look here,"},{"Start":"04:20.810 ","End":"04:24.890","Text":"that\u0027s exactly what we were trying to show, and so we\u0027re done."},{"Start":"04:24.890 ","End":"04:27.480","Text":"That concludes this exercise."}],"ID":26632},{"Watched":false,"Name":"Exercise 10","Duration":"3m 12s","ChapterTopicVideoID":25829,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.090 ","End":"00:07.060","Text":"In this exercise, we have to prove that 4^n minus 1 is divisible by"},{"Start":"00:07.060 ","End":"00:15.135","Text":"15 for every even n. Now this is not for every n,"},{"Start":"00:15.135 ","End":"00:19.620","Text":"so we\u0027re going to have to modify the proof by induction."},{"Start":"00:19.620 ","End":"00:27.355","Text":"We\u0027ll denote P of n to be the statement that 4^n minus 1 is divisible by 15."},{"Start":"00:27.355 ","End":"00:33.490","Text":"Our plan is to prove that P of n is true for all n. Here\u0027s how we modify the induction."},{"Start":"00:33.490 ","End":"00:34.995","Text":"In step 1,"},{"Start":"00:34.995 ","End":"00:38.555","Text":"we check P of 2 not P of 1."},{"Start":"00:38.555 ","End":"00:42.505","Text":"You want the first even number, which is 2."},{"Start":"00:42.505 ","End":"00:48.650","Text":"Step 2 would be to assume that P of n is true for a particular n,"},{"Start":"00:48.650 ","End":"00:52.055","Text":"but an even n may or may not need that fact."},{"Start":"00:52.055 ","End":"00:57.530","Text":"From this, we\u0027ll prove that P of n plus 2 is true for this"},{"Start":"00:57.530 ","End":"01:03.810","Text":"n. The difference is that we don\u0027t start from 1,"},{"Start":"01:03.810 ","End":"01:04.995","Text":"we start from 2,"},{"Start":"01:04.995 ","End":"01:07.140","Text":"and we go in jumps of 2."},{"Start":"01:07.140 ","End":"01:09.210","Text":"Now why does this work?"},{"Start":"01:09.210 ","End":"01:13.350","Text":"P of 2 is true directly from step 1,"},{"Start":"01:13.350 ","End":"01:16.870","Text":"and then from 2 we get 3."},{"Start":"01:16.870 ","End":"01:18.440","Text":"In other words, if it\u0027s true for n,"},{"Start":"01:18.440 ","End":"01:19.670","Text":"it\u0027s true for n plus 2."},{"Start":"01:19.670 ","End":"01:22.070","Text":"From 2, we get to 4, from 4,"},{"Start":"01:22.070 ","End":"01:23.100","Text":"we get to 6,"},{"Start":"01:23.100 ","End":"01:25.400","Text":"from 6, we get to 8."},{"Start":"01:25.400 ","End":"01:26.765","Text":"If we keep going,"},{"Start":"01:26.765 ","End":"01:29.000","Text":"we\u0027ll reach any even number."},{"Start":"01:29.000 ","End":"01:30.635","Text":"That\u0027s the rationale."},{"Start":"01:30.635 ","End":"01:32.840","Text":"Now let\u0027s do these 3 steps."},{"Start":"01:32.840 ","End":"01:39.870","Text":"First of all, P of 2 says that 4 squared minus 1 is divisible by 15."},{"Start":"01:39.870 ","End":"01:43.090","Text":"Well, it is 15,15 is divisible by 15,"},{"Start":"01:43.090 ","End":"01:44.990","Text":"so we\u0027re okay there."},{"Start":"01:44.990 ","End":"01:51.170","Text":"Now we\u0027re going to assume the induction hypothesis that for a particular n,"},{"Start":"01:51.170 ","End":"01:54.425","Text":"4^n minus 1 is divisible by 15."},{"Start":"01:54.425 ","End":"01:58.340","Text":"What we\u0027re going to prove from this is that P of n"},{"Start":"01:58.340 ","End":"02:03.030","Text":"plus 2 is true for this n. What is that mean?"},{"Start":"02:03.030 ","End":"02:08.070","Text":"It means that 4^n plus 2 minus 1 is divisible by 15."},{"Start":"02:08.070 ","End":"02:12.930","Text":"We\u0027re going to use this to prove this."},{"Start":"02:12.930 ","End":"02:16.250","Text":"Here\u0027s how we do it. Start with this expression,"},{"Start":"02:16.250 ","End":"02:18.575","Text":"4^n plus 2 minus 1."},{"Start":"02:18.575 ","End":"02:20.240","Text":"We can write this as 4 squared,"},{"Start":"02:20.240 ","End":"02:23.195","Text":"4^n, and 4 squared is 16,"},{"Start":"02:23.195 ","End":"02:25.940","Text":"so we have 16 times 4^n minus 1,"},{"Start":"02:25.940 ","End":"02:30.680","Text":"and 16 is 15 plus 1."},{"Start":"02:30.680 ","End":"02:34.220","Text":"Yeah, I should have said 16 is 15 plus 1."},{"Start":"02:34.220 ","End":"02:39.650","Text":"We can break this term up into 15 times 4^n plus 1 times 4^n."},{"Start":"02:39.650 ","End":"02:43.680","Text":"We don\u0027t need to write the 1 and the minus 1 from here."},{"Start":"02:43.680 ","End":"02:46.565","Text":"Now, we have 2 parts here,"},{"Start":"02:46.565 ","End":"02:49.625","Text":"and clearly each of them is divisible by 15."},{"Start":"02:49.625 ","End":"02:52.670","Text":"This is divisible by 15 because it\u0027s 15 times something,"},{"Start":"02:52.670 ","End":"02:57.170","Text":"and this is divisible by 15 from the induction hypothesis."},{"Start":"02:57.170 ","End":"03:06.735","Text":"The sum of divisible by 15 is also 4^n plus 2 minus 1 divisible by 15."},{"Start":"03:06.735 ","End":"03:09.680","Text":"That\u0027s the induction step prove done."},{"Start":"03:09.680 ","End":"03:12.510","Text":"That concludes this exercise."}],"ID":26633},{"Watched":false,"Name":"Exercise 11","Duration":"2m 29s","ChapterTopicVideoID":25830,"CourseChapterTopicPlaylistID":246309,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this exercise, it\u0027s assumed that you have some knowledge"},{"Start":"00:03.750 ","End":"00:07.260","Text":"of matrices that\u0027s in linear algebra,"},{"Start":"00:07.260 ","End":"00:13.245","Text":"and in particular, you know what a 2-by-2 matrix is and how to multiply matrices."},{"Start":"00:13.245 ","End":"00:15.990","Text":"If not, you just skip this exercise."},{"Start":"00:15.990 ","End":"00:19.740","Text":"We have to prove by induction that if we take the matrix"},{"Start":"00:19.740 ","End":"00:23.955","Text":"0 0 0 a and raise it to the power of n,"},{"Start":"00:23.955 ","End":"00:27.675","Text":"we get 0 0 0 a^n."},{"Start":"00:27.675 ","End":"00:30.825","Text":"It\u0027s like putting the power of n just down here."},{"Start":"00:30.825 ","End":"00:35.760","Text":"This works for all natural numbers n. By the way,"},{"Start":"00:35.760 ","End":"00:37.740","Text":"it doesn\u0027t really matter what the field is."},{"Start":"00:37.740 ","End":"00:40.290","Text":"We can assume that it\u0027s the real numbers,"},{"Start":"00:40.290 ","End":"00:42.430","Text":"but it works over any field,"},{"Start":"00:42.430 ","End":"00:44.580","Text":"but we\u0027ll take it as the real number,"},{"Start":"00:44.580 ","End":"00:46.635","Text":"so a is a real number."},{"Start":"00:46.635 ","End":"00:52.925","Text":"Now let P of n be the statement that this is true for a particular"},{"Start":"00:52.925 ","End":"00:59.315","Text":"n. We have to prove that P of n is true for all n. We\u0027re going to do it by induction,"},{"Start":"00:59.315 ","End":"01:02.150","Text":"so the first step is the base case to show that it\u0027s"},{"Start":"01:02.150 ","End":"01:06.420","Text":"true and n equals 1 that this is equal to this."},{"Start":"01:06.700 ","End":"01:10.910","Text":"That\u0027s clearly true because to the power of 1 is itself"},{"Start":"01:10.910 ","End":"01:14.800","Text":"and a^1 is also a itself, so that\u0027s trivial."},{"Start":"01:14.800 ","End":"01:19.129","Text":"Now we\u0027re going to assume that for a particular end that this holds,"},{"Start":"01:19.129 ","End":"01:24.440","Text":"this is our induction hypothesis and I\u0027m going to use this to"},{"Start":"01:24.440 ","End":"01:30.870","Text":"prove that P of n plus 1 is true which means that if we put n plus 1 here and here,"},{"Start":"01:30.870 ","End":"01:33.390","Text":"this will also be true."},{"Start":"01:34.450 ","End":"01:38.235","Text":"0 0 0 a^n plus 1."},{"Start":"01:38.235 ","End":"01:41.930","Text":"In matrices, we also have a rule of exponents and it\u0027s"},{"Start":"01:41.930 ","End":"01:45.410","Text":"this to the power of 1 times this to the power of"},{"Start":"01:45.410 ","End":"01:53.450","Text":"n. Now we can use the induction hypothesis on this part and say that this is equal to,"},{"Start":"01:53.450 ","End":"01:56.075","Text":"just put the n down here."},{"Start":"01:56.075 ","End":"01:58.595","Text":"That\u0027s our induction hypothesis."},{"Start":"01:58.595 ","End":"02:02.450","Text":"Now we have to multiply these 2 matrices. Let\u0027s see."},{"Start":"02:02.450 ","End":"02:05.750","Text":"This row with this column gives 0,"},{"Start":"02:05.750 ","End":"02:08.570","Text":"this row with this column also 0."},{"Start":"02:08.570 ","End":"02:10.580","Text":"This row with this column,"},{"Start":"02:10.580 ","End":"02:13.325","Text":"0 times 0 plus a times 0 is 0,"},{"Start":"02:13.325 ","End":"02:19.000","Text":"and this with this gives us 0 times 0 plus a times a^n."},{"Start":"02:19.000 ","End":"02:23.215","Text":"Really just a times a^n,"},{"Start":"02:23.215 ","End":"02:26.655","Text":"which is a^n plus 1."},{"Start":"02:26.655 ","End":"02:28.560","Text":"That\u0027s what we want to prove,"},{"Start":"02:28.560 ","End":"02:30.790","Text":"and so we\u0027re done."}],"ID":26634}],"Thumbnail":null,"ID":246309},{"Name":"Summation and Sigma notation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"The Sigma Notation for Summation","Duration":"6m 55s","ChapterTopicVideoID":25842,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In this clip, I\u0027m going to introduce"},{"Start":"00:03.045 ","End":"00:08.055","Text":"the Sigma notation for summation in case you haven\u0027t seen it already."},{"Start":"00:08.055 ","End":"00:11.475","Text":"This is the Greek letter Sigma."},{"Start":"00:11.475 ","End":"00:13.860","Text":"It\u0027s capital Sigma, actually."},{"Start":"00:13.860 ","End":"00:18.790","Text":"There\u0027s also a small sigma used in statistics, for example."},{"Start":"00:18.800 ","End":"00:21.105","Text":"What is this thing?"},{"Start":"00:21.105 ","End":"00:22.470","Text":"I\u0027ll give you an example."},{"Start":"00:22.470 ","End":"00:27.000","Text":"Suppose you have a sum of several terms of the form,"},{"Start":"00:27.000 ","End":"00:31.590","Text":"1 squared plus 2 squared plus 3 squared plus"},{"Start":"00:31.590 ","End":"00:36.765","Text":"4 squared plus 5 squared plus 6 squared plus 7 squared,"},{"Start":"00:36.765 ","End":"00:39.735","Text":"and I\u0027ll stop at 7 but it could have even been longer."},{"Start":"00:39.735 ","End":"00:44.150","Text":"Now, this sum is quite tiresome to write"},{"Start":"00:44.150 ","End":"00:49.055","Text":"out in full so we want a shorthand way of writing this."},{"Start":"00:49.055 ","End":"00:58.145","Text":"If you notice, all of them are of the form n squared where n is some number,"},{"Start":"00:58.145 ","End":"01:00.605","Text":"1, 2, 3, 4, 5, 6, or 7."},{"Start":"01:00.605 ","End":"01:02.840","Text":"In fact, they\u0027re even in sequence."},{"Start":"01:02.840 ","End":"01:05.540","Text":"The way we write this is a convention."},{"Start":"01:05.540 ","End":"01:11.655","Text":"If write it as Sigma and we say the sum of n squared,"},{"Start":"01:11.655 ","End":"01:13.820","Text":"and here and here,"},{"Start":"01:13.820 ","End":"01:16.175","Text":"we write where n goes from and to."},{"Start":"01:16.175 ","End":"01:18.650","Text":"From n equals 1,"},{"Start":"01:18.650 ","End":"01:20.840","Text":"2, n equals 7."},{"Start":"01:20.840 ","End":"01:23.330","Text":"But we don\u0027t write the n equals at the top."},{"Start":"01:23.330 ","End":"01:25.910","Text":"n goes from 1 to 7,"},{"Start":"01:25.910 ","End":"01:27.695","Text":"the sum of n squared,"},{"Start":"01:27.695 ","End":"01:29.840","Text":"which means you let n equals 1,"},{"Start":"01:29.840 ","End":"01:32.820","Text":"n equals 2, and so on."},{"Start":"01:33.560 ","End":"01:36.425","Text":"Now, another example,"},{"Start":"01:36.425 ","End":"01:42.365","Text":"1 cubed plus 2 cubed plus 3 cubed."},{"Start":"01:42.365 ","End":"01:44.675","Text":"I think you\u0027ve probably got the idea."},{"Start":"01:44.675 ","End":"01:49.340","Text":"The general pattern is n cubed and n goes from 1 to 3,"},{"Start":"01:49.340 ","End":"01:57.165","Text":"so we write this as Sigma n equals 1 to 3 of n cubed."},{"Start":"01:57.165 ","End":"02:00.150","Text":"Now, let\u0027s try a reverse example."},{"Start":"02:00.150 ","End":"02:05.490","Text":"Let\u0027s start with the Sigma n equals 4"},{"Start":"02:05.490 ","End":"02:13.170","Text":"to 8 of 2n."},{"Start":"02:13.170 ","End":"02:16.570","Text":"What we have to do is substitute n equals 4,"},{"Start":"02:16.570 ","End":"02:17.930","Text":"then 5, and 6, and 7,"},{"Start":"02:17.930 ","End":"02:19.990","Text":"then 8 in this expression."},{"Start":"02:19.990 ","End":"02:23.460","Text":"For each of these, we put an addition sign between."},{"Start":"02:23.460 ","End":"02:25.924","Text":"What I\u0027m saying is it\u0027s 2 times 4,"},{"Start":"02:25.924 ","End":"02:27.215","Text":"where n equals 4,"},{"Start":"02:27.215 ","End":"02:29.620","Text":"then I let n equals 5."},{"Start":"02:29.620 ","End":"02:31.680","Text":"All along, I\u0027m adding,"},{"Start":"02:31.680 ","End":"02:33.195","Text":"Sigma is the sum."},{"Start":"02:33.195 ","End":"02:35.100","Text":"Then 2 times 6,"},{"Start":"02:35.100 ","End":"02:36.765","Text":"2 times 7,"},{"Start":"02:36.765 ","End":"02:38.795","Text":"and 2 times 8."},{"Start":"02:38.795 ","End":"02:43.760","Text":"There also is a numerical answer but I\u0027m not bothered to actually compute it."},{"Start":"02:43.760 ","End":"02:50.040","Text":"Another example, the sum from n equals"},{"Start":"02:50.040 ","End":"02:59.430","Text":"2 to 5 of minus 1 to the power of n times n plus 1."},{"Start":"02:59.430 ","End":"03:02.395","Text":"Well, let\u0027s see what this equals."},{"Start":"03:02.395 ","End":"03:10.540","Text":"I first let n equals 2 minus 1 squared is 1 and 2 plus 1 is 3,"},{"Start":"03:10.540 ","End":"03:12.830","Text":"so this becomes 3."},{"Start":"03:12.830 ","End":"03:19.085","Text":"Next term, n equals 3 minus 1 to the power of 3 is minus 1,"},{"Start":"03:19.085 ","End":"03:20.435","Text":"and here, it\u0027s 4,"},{"Start":"03:20.435 ","End":"03:22.745","Text":"so we get minus 4."},{"Start":"03:22.745 ","End":"03:24.935","Text":"Then when n is 4,"},{"Start":"03:24.935 ","End":"03:28.325","Text":"again, we get plus here, and here, we get 5."},{"Start":"03:28.325 ","End":"03:31.770","Text":"When n is 5,"},{"Start":"03:31.770 ","End":"03:34.835","Text":"we get minus 6."},{"Start":"03:34.835 ","End":"03:37.660","Text":"The thing that makes this go plus, minus, plus,"},{"Start":"03:37.660 ","End":"03:41.040","Text":"minus is the minus 1 to the n,"},{"Start":"03:41.040 ","End":"03:42.480","Text":"and we\u0027ll see this a lot."},{"Start":"03:42.480 ","End":"03:46.490","Text":"In fact, this thing is called an alternating series."},{"Start":"03:46.490 ","End":"03:48.880","Text":"In case you don\u0027t know what a series is,"},{"Start":"03:48.880 ","End":"03:50.830","Text":"each of these things is a series,"},{"Start":"03:50.830 ","End":"03:55.120","Text":"it\u0027s a bunch of quantity with pluses in the middle."},{"Start":"03:55.120 ","End":"03:57.520","Text":"If it\u0027s commas it\u0027s a sequence,"},{"Start":"03:57.520 ","End":"03:59.750","Text":"and if it has pluses, it\u0027s a series."},{"Start":"03:59.750 ","End":"04:02.445","Text":"This is a series, this is a series, this is a series,"},{"Start":"04:02.445 ","End":"04:05.350","Text":"this is a series, and this one is alternating."},{"Start":"04:05.350 ","End":"04:09.725","Text":"I\u0027d like you to note that a series is allowed to be infinite."},{"Start":"04:09.725 ","End":"04:17.103","Text":"For example, we could have the sum from n equals 1"},{"Start":"04:17.103 ","End":"04:23.960","Text":"to infinity of 1 over n. What this is equal to,"},{"Start":"04:23.960 ","End":"04:28.040","Text":"if n equals 1, then it\u0027s 1 over 1."},{"Start":"04:28.040 ","End":"04:30.005","Text":"If n equals 2,"},{"Start":"04:30.005 ","End":"04:31.715","Text":"it\u0027s 1 over 2."},{"Start":"04:31.715 ","End":"04:34.200","Text":"If n is 3,"},{"Start":"04:34.250 ","End":"04:37.035","Text":"it\u0027s 1 over 3,"},{"Start":"04:37.035 ","End":"04:41.830","Text":"1 over 4, and so on."},{"Start":"04:41.960 ","End":"04:44.930","Text":"Sometimes, I put the general term,"},{"Start":"04:44.930 ","End":"04:48.985","Text":"1 over n dot, dot, dot."},{"Start":"04:48.985 ","End":"04:54.215","Text":"Now, let\u0027s look at 1 final example."},{"Start":"04:54.215 ","End":"05:02.720","Text":"There is a mathematical theorem that says that e to the power of x is equal to"},{"Start":"05:02.720 ","End":"05:07.520","Text":"the sum n goes from 0 to"},{"Start":"05:07.520 ","End":"05:15.815","Text":"infinity of x to the power of n over n factorial."},{"Start":"05:15.815 ","End":"05:20.450","Text":"Let\u0027s expand this and see what this is."},{"Start":"05:20.450 ","End":"05:24.845","Text":"This will give us what we call the infinite series for e to the x."},{"Start":"05:24.845 ","End":"05:29.860","Text":"First, n equals 0. x to the 0 is 1,"},{"Start":"05:29.860 ","End":"05:34.950","Text":"and 0 factorial by convention is 1 also."},{"Start":"05:34.950 ","End":"05:37.230","Text":"This gives us 1."},{"Start":"05:37.230 ","End":"05:39.330","Text":"When n equals 1,"},{"Start":"05:39.330 ","End":"05:41.175","Text":"this gives us x here,"},{"Start":"05:41.175 ","End":"05:45.939","Text":"1 factorial is 1 so this is x."},{"Start":"05:46.640 ","End":"05:48.960","Text":"Then if n equals 2,"},{"Start":"05:48.960 ","End":"05:54.210","Text":"we get x squared over 2 factorial,"},{"Start":"05:54.210 ","End":"05:57.945","Text":"and I\u0027ll leave it as 2 factorial even though I could compute that,"},{"Start":"05:57.945 ","End":"05:59.675","Text":"and you get the idea."},{"Start":"05:59.675 ","End":"06:07.139","Text":"x to the power of 3 over 3 factorial and so on and so on."},{"Start":"06:07.139 ","End":"06:12.010","Text":"The general term is x to the n over n factorial,"},{"Start":"06:12.010 ","End":"06:13.555","Text":"I\u0027ll just copy that from here."},{"Start":"06:13.555 ","End":"06:15.845","Text":"But it goes on forever."},{"Start":"06:15.845 ","End":"06:22.870","Text":"Some people even write e to the x in this form instead of the short Sigma notation."},{"Start":"06:22.870 ","End":"06:26.630","Text":"This is somewhat cumbersome and tedious."},{"Start":"06:26.630 ","End":"06:28.595","Text":"For example, if you wanted e to the 4x,"},{"Start":"06:28.595 ","End":"06:31.310","Text":"you\u0027d have to say 1 plus 4x plus 4x,"},{"Start":"06:31.310 ","End":"06:33.970","Text":"all squared, and so on."},{"Start":"06:33.970 ","End":"06:38.510","Text":"This notation does have its advantages but in general,"},{"Start":"06:38.510 ","End":"06:42.215","Text":"we\u0027ll be using the Sigma notation which is"},{"Start":"06:42.215 ","End":"06:47.250","Text":"widely accepted in mathematics everywhere in the world."},{"Start":"06:48.320 ","End":"06:51.930","Text":"Now, you know what it is."},{"Start":"06:51.930 ","End":"06:55.780","Text":"Done with this introduction."}],"ID":26646},{"Watched":false,"Name":"Exercise 1","Duration":"3m 24s","ChapterTopicVideoID":25835,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.225","Text":"This exercise, it\u0027s really 9 exercises and each of them we have a Sigma,"},{"Start":"00:06.225 ","End":"00:09.490","Text":"a sum and we just have to expand it to interpret it."},{"Start":"00:09.490 ","End":"00:11.785","Text":"We don\u0027t need to actually get the final answer."},{"Start":"00:11.785 ","End":"00:14.200","Text":"Once we do the first 1, you\u0027ll get the idea."},{"Start":"00:14.200 ","End":"00:20.310","Text":"Let\u0027s see, the sum as n goes from 0 to 10 of 4^n."},{"Start":"00:20.310 ","End":"00:24.520","Text":"What we do is we successively substitute n equals 0,"},{"Start":"00:24.520 ","End":"00:26.110","Text":"n equals 1, n equals 2,"},{"Start":"00:26.110 ","End":"00:29.630","Text":"and so on and we add the terms that we get."},{"Start":"00:29.630 ","End":"00:33.310","Text":"We end up with 4^0, 4^1, 2,"},{"Start":"00:33.310 ","End":"00:34.900","Text":"3, 4, 5, 6, 7,"},{"Start":"00:34.900 ","End":"00:37.340","Text":"8, 9, 10 and that\u0027s it."},{"Start":"00:37.340 ","End":"00:44.160","Text":"Part B, the sum k goes from 1 to 4 of 2k."},{"Start":"00:44.160 ","End":"00:46.110","Text":"We let k equal 1 and 2,"},{"Start":"00:46.110 ","End":"00:47.850","Text":"then 3, then 4."},{"Start":"00:47.850 ","End":"00:49.920","Text":"It\u0027s 2 times 1, 2 times 2,"},{"Start":"00:49.920 ","End":"00:51.855","Text":"2 times 3, 2 times 4."},{"Start":"00:51.855 ","End":"00:54.325","Text":"That\u0027s it, we don\u0027t have to go any further."},{"Start":"00:54.325 ","End":"00:56.090","Text":"Here, once again,"},{"Start":"00:56.090 ","End":"00:58.760","Text":"we\u0027re back to n. Notice that the letters vary."},{"Start":"00:58.760 ","End":"01:01.175","Text":"See we have an i and a t and so on."},{"Start":"01:01.175 ","End":"01:02.465","Text":"n times a_n,"},{"Start":"01:02.465 ","End":"01:05.505","Text":"n goes from 4-10."},{"Start":"01:05.505 ","End":"01:08.205","Text":"Only assume that an is some sequence."},{"Start":"01:08.205 ","End":"01:12.690","Text":"Let n equal 4, we get 4a_4 and so on 5a_5,"},{"Start":"01:12.690 ","End":"01:15.750","Text":"6a_6 up to 10a_10."},{"Start":"01:15.750 ","End":"01:19.515","Text":"Here we\u0027re going from 7 to 11 with i."},{"Start":"01:19.515 ","End":"01:23.190","Text":"When i is 7, we get 4, 7 squared a_7."},{"Start":"01:23.190 ","End":"01:26.145","Text":"Similarly for 8, 9, 10, and 11."},{"Start":"01:26.145 ","End":"01:31.365","Text":"The next 1 we used the index t. t goes from 1 to 8,"},{"Start":"01:31.365 ","End":"01:34.575","Text":"the sum of tx^t."},{"Start":"01:34.575 ","End":"01:38.760","Text":"When t is 1, we get 1x^1, and so on,"},{"Start":"01:38.760 ","End":"01:44.295","Text":"up to 8x^8. What do we have here?"},{"Start":"01:44.295 ","End":"01:46.375","Text":"k goes from 4 to 10."},{"Start":"01:46.375 ","End":"01:49.955","Text":"Notice that there\u0027s a k here and there\u0027s an n here."},{"Start":"01:49.955 ","End":"01:54.890","Text":"The index is k, n is just some constant that stays."},{"Start":"01:54.890 ","End":"01:57.315","Text":"We get na_5."},{"Start":"01:57.315 ","End":"01:59.445","Text":"The 5 is from 4 plus 1."},{"Start":"01:59.445 ","End":"02:02.380","Text":"Then we have 5 plus 1 which is 6,"},{"Start":"02:02.380 ","End":"02:05.315","Text":"up to 10 plus 1, which is 11."},{"Start":"02:05.315 ","End":"02:09.275","Text":"But the end stays n because k is the index."},{"Start":"02:09.275 ","End":"02:13.350","Text":"Here also, another trick to try and fool you."},{"Start":"02:13.350 ","End":"02:18.105","Text":"k is the index goes from 1 to 10, 4n doesn\u0027t change."},{"Start":"02:18.105 ","End":"02:22.425","Text":"We get 4n plus 4n plus 4n."},{"Start":"02:22.425 ","End":"02:27.170","Text":"The term doesn\u0027t depend on k. Still we need to count."},{"Start":"02:27.170 ","End":"02:30.275","Text":"That\u0027s why I\u0027ve put these numbers here in faint."},{"Start":"02:30.275 ","End":"02:32.970","Text":"1, 2, 3, we have to count 4,"},{"Start":"02:32.970 ","End":"02:35.745","Text":"5, 6, 7, 8, 9, 10."},{"Start":"02:35.745 ","End":"02:37.320","Text":"So k is 1, k is 2,"},{"Start":"02:37.320 ","End":"02:40.680","Text":"k is 3, whatever k is, it\u0027s 4n."},{"Start":"02:40.680 ","End":"02:43.970","Text":"The sum of k squared plus 1,"},{"Start":"02:43.970 ","End":"02:46.790","Text":"where k goes from minus 1 to 3."},{"Start":"02:46.790 ","End":"02:48.440","Text":"We let k equals minus 1 and 0,"},{"Start":"02:48.440 ","End":"02:49.610","Text":"then 1, then 2,"},{"Start":"02:49.610 ","End":"02:54.990","Text":"then 3 minus 1 squared plus 1 and 0 squared plus 1,"},{"Start":"02:54.990 ","End":"02:58.290","Text":"1 squared plus 1, and so on up to 3 squared plus 1."},{"Start":"02:58.290 ","End":"03:03.330","Text":"The last 1 is l goes from 1 to 3 and we have l here,"},{"Start":"03:03.330 ","End":"03:06.330","Text":"and we have l in the subscript here."},{"Start":"03:06.330 ","End":"03:11.625","Text":"When l is 1, it\u0027s 1 squared and here is x 2 times x1 is x_2."},{"Start":"03:11.625 ","End":"03:16.815","Text":"When l is 2, we get x_4 because that\u0027s 20 L and 2 squared."},{"Start":"03:16.815 ","End":"03:22.335","Text":"When l is 3, we get x_6 here and it\u0027s 3 squared and this is what we get."},{"Start":"03:22.335 ","End":"03:25.360","Text":"That concludes this exercise."}],"ID":26639},{"Watched":false,"Name":"Exercise 2","Duration":"8m 53s","ChapterTopicVideoID":25836,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"This exercise has 10 parts,"},{"Start":"00:03.075 ","End":"00:04.650","Text":"and in each of them,"},{"Start":"00:04.650 ","End":"00:10.184","Text":"we have to write the sums using the summation sign, sigma."},{"Start":"00:10.184 ","End":"00:13.365","Text":"Usually, there\u0027s more than 1 answer possible,"},{"Start":"00:13.365 ","End":"00:16.845","Text":"I just chose the one that seems most straight forward."},{"Start":"00:16.845 ","End":"00:18.795","Text":"Let\u0027s start with a,"},{"Start":"00:18.795 ","End":"00:22.125","Text":"on a new page, 1 plus 2 plus 4,"},{"Start":"00:22.125 ","End":"00:24.480","Text":"and so on up to 128."},{"Start":"00:24.480 ","End":"00:28.500","Text":"You immediately recognize these as powers of 2 so"},{"Start":"00:28.500 ","End":"00:33.060","Text":"we can write this as 2 to the 0 plus 2 to the 1 and so on."},{"Start":"00:33.060 ","End":"00:35.280","Text":"Here it\u0027s 2 to the 7th."},{"Start":"00:35.280 ","End":"00:38.450","Text":"You don\u0027t even have to compute this with logarithms."},{"Start":"00:38.450 ","End":"00:41.000","Text":"You just keep counting, 0, 1, 2,"},{"Start":"00:41.000 ","End":"00:42.410","Text":"3, 4, 5, 6,"},{"Start":"00:42.410 ","End":"00:43.960","Text":"7, and it\u0027s going to be right."},{"Start":"00:43.960 ","End":"00:47.690","Text":"It looks like it\u0027s 2 to the power of a moving index,"},{"Start":"00:47.690 ","End":"00:50.075","Text":"and the index will go from 0 to 7."},{"Start":"00:50.075 ","End":"00:54.170","Text":"Let\u0027s choose the letter k so k will go from 0 to 7,"},{"Start":"00:54.170 ","End":"00:56.865","Text":"2 to the k, and that\u0027s it."},{"Start":"00:56.865 ","End":"01:00.840","Text":"Part b, 2 plus 4 plus 6 and so on."},{"Start":"01:00.840 ","End":"01:04.385","Text":"The pattern looks like they\u0027re even numbers."},{"Start":"01:04.385 ","End":"01:06.740","Text":"Since these are even, we can take 2 out."},{"Start":"01:06.740 ","End":"01:08.180","Text":"We can write this as 2 times 1,"},{"Start":"01:08.180 ","End":"01:11.795","Text":"2 times 2, up to 2 times 10."},{"Start":"01:11.795 ","End":"01:15.320","Text":"It\u0027s 2 times something that goes from 1 to 10,"},{"Start":"01:15.320 ","End":"01:17.515","Text":"we use another k again."},{"Start":"01:17.515 ","End":"01:19.860","Text":"K goes from 1 to 10,"},{"Start":"01:19.860 ","End":"01:23.095","Text":"2 times k. The next one,"},{"Start":"01:23.095 ","End":"01:25.730","Text":"looks like the odd numbers 1,"},{"Start":"01:25.730 ","End":"01:27.830","Text":"3, 5, up to 19."},{"Start":"01:27.830 ","End":"01:34.760","Text":"We can write these in terms of these because these numbers are 1 less than these numbers."},{"Start":"01:34.760 ","End":"01:37.190","Text":"One approach would be to write it as 2 minus 1,"},{"Start":"01:37.190 ","End":"01:40.325","Text":"4 minus 1, 6 minus 1 plus etc,"},{"Start":"01:40.325 ","End":"01:45.150","Text":"up to 20 minus 1 and then as before,"},{"Start":"01:45.150 ","End":"01:46.350","Text":"we can break 2, 4,"},{"Start":"01:46.350 ","End":"01:47.910","Text":"6, 2 times 1,"},{"Start":"01:47.910 ","End":"01:49.815","Text":"2 times 2, 2 times 3."},{"Start":"01:49.815 ","End":"01:56.905","Text":"If this is the k and what we have is 2k minus 1 from 1 to 10."},{"Start":"01:56.905 ","End":"01:59.555","Text":"That\u0027s not the only way of doing it."},{"Start":"01:59.555 ","End":"02:01.580","Text":"Instead of 2 minus 1,"},{"Start":"02:01.580 ","End":"02:03.305","Text":"4 minus 1, and so on,"},{"Start":"02:03.305 ","End":"02:05.240","Text":"we can look at it as 0 plus 1,"},{"Start":"02:05.240 ","End":"02:06.485","Text":"2 plus 1,"},{"Start":"02:06.485 ","End":"02:08.825","Text":"and end with 18 plus one."},{"Start":"02:08.825 ","End":"02:12.320","Text":"In this case, you can see that the sum would be written as"},{"Start":"02:12.320 ","End":"02:16.715","Text":"k goes from 0 to 9 of 2k plus 1."},{"Start":"02:16.715 ","End":"02:19.530","Text":"Actually, I should have put brackets here,"},{"Start":"02:19.530 ","End":"02:21.780","Text":"so it\u0027s more precise,"},{"Start":"02:21.780 ","End":"02:25.890","Text":"otherwise you might think it\u0027s the sum of 2 k and then minus 1 so yeah,"},{"Start":"02:25.890 ","End":"02:29.415","Text":"better put some brackets here and here."},{"Start":"02:29.415 ","End":"02:32.250","Text":"Okay, let\u0027s go onto the next one."},{"Start":"02:32.250 ","End":"02:35.840","Text":"This one\u0027s a bit more complicated but if you look at it,"},{"Start":"02:35.840 ","End":"02:38.570","Text":"we have a sum of products, and in each case,"},{"Start":"02:38.570 ","End":"02:42.530","Text":"the number on the left is a consecutive number, 1,"},{"Start":"02:42.530 ","End":"02:47.380","Text":"2, 3, 4, 5, 6, 7."},{"Start":"02:47.380 ","End":"02:50.695","Text":"The left part could be written as k,"},{"Start":"02:50.695 ","End":"02:52.960","Text":"which goes from 1 to 7."},{"Start":"02:52.960 ","End":"02:56.090","Text":"Now the right side is just 1 more than the left."},{"Start":"02:56.090 ","End":"02:57.720","Text":"1 plus 1 is 2,"},{"Start":"02:57.720 ","End":"02:59.580","Text":"2 plus 1 is 3."},{"Start":"02:59.580 ","End":"03:03.490","Text":"If this is k, this would be k plus 1, k plus 1,"},{"Start":"03:03.490 ","End":"03:05.905","Text":"3 plus 1, 4 plus 1,"},{"Start":"03:05.905 ","End":"03:09.125","Text":"5 plus 1, 6 plus 1, 7 plus 1."},{"Start":"03:09.125 ","End":"03:12.555","Text":"Altogether, it comes out like so."},{"Start":"03:12.555 ","End":"03:14.055","Text":"Okay, next 1."},{"Start":"03:14.055 ","End":"03:15.465","Text":"Little bit different."},{"Start":"03:15.465 ","End":"03:19.940","Text":"Looks like we also have a number and its successor,"},{"Start":"03:19.940 ","End":"03:21.969","Text":"but this time there are gaps."},{"Start":"03:21.969 ","End":"03:24.430","Text":"We have on the left, 1,"},{"Start":"03:24.430 ","End":"03:29.995","Text":"3, 5, 7 and so on. We have odd numbers,"},{"Start":"03:29.995 ","End":"03:32.495","Text":"and then the one on the right is one more."},{"Start":"03:32.495 ","End":"03:36.695","Text":"Might be easier to start with the ones on the right with 2,"},{"Start":"03:36.695 ","End":"03:39.770","Text":"4, 6, 8 and so on,"},{"Start":"03:39.770 ","End":"03:42.070","Text":"because that would be 2k,"},{"Start":"03:42.070 ","End":"03:47.570","Text":"when k goes from 1 to 22 and then the left one you could get by subtracting 1."},{"Start":"03:47.570 ","End":"03:52.655","Text":"What I\u0027m suggesting is k goes from 1 to 20 to the right side is 2k,"},{"Start":"03:52.655 ","End":"03:55.840","Text":"and the left side will be 2k minus 1."},{"Start":"03:55.840 ","End":"03:57.480","Text":"This is what we\u0027ll get."},{"Start":"03:57.480 ","End":"03:59.450","Text":"Now on to the next one."},{"Start":"03:59.450 ","End":"04:02.120","Text":"Let\u0027s see if we can identify the pattern."},{"Start":"04:02.120 ","End":"04:06.620","Text":"On the left, it looks like 3, 6, 9, 12,"},{"Start":"04:06.620 ","End":"04:09.185","Text":"numbers divisible by 3."},{"Start":"04:09.185 ","End":"04:11.450","Text":"That looks like 3k, 3 times 1,"},{"Start":"04:11.450 ","End":"04:13.980","Text":"3 times 2, 3 times 3,"},{"Start":"04:13.980 ","End":"04:15.840","Text":"up to 3 times 7."},{"Start":"04:15.840 ","End":"04:18.035","Text":"What about the number on the right?"},{"Start":"04:18.035 ","End":"04:21.380","Text":"I think the pattern is that if you add 1 to 1,"},{"Start":"04:21.380 ","End":"04:24.560","Text":"you get 2, 2 plus 1 is 3,"},{"Start":"04:24.560 ","End":"04:26.540","Text":"3 plus 1 is 4,"},{"Start":"04:26.540 ","End":"04:28.130","Text":"4 plus 1 is 5,"},{"Start":"04:28.130 ","End":"04:29.570","Text":"5 plus 1 is 6,"},{"Start":"04:29.570 ","End":"04:31.300","Text":"6 plus 1 is 7,"},{"Start":"04:31.300 ","End":"04:33.925","Text":"7 plus 1 is 8."},{"Start":"04:33.925 ","End":"04:37.970","Text":"Number on the left is the 3k part,"},{"Start":"04:37.970 ","End":"04:40.490","Text":"and on the right it\u0027s k plus 1,"},{"Start":"04:40.490 ","End":"04:42.905","Text":"because we take the k and add 1."},{"Start":"04:42.905 ","End":"04:49.030","Text":"We have sum k goes from 1 to 7 of 3k times k plus 1."},{"Start":"04:49.030 ","End":"04:54.605","Text":"Next, 5 squared plus 7 squared and so on up to 27 squared."},{"Start":"04:54.605 ","End":"04:56.060","Text":"Here we have an ellipsis,"},{"Start":"04:56.060 ","End":"04:57.260","Text":"dot dot dot,"},{"Start":"04:57.260 ","End":"05:00.190","Text":"and they assume that the pattern is clear and I think it is."},{"Start":"05:00.190 ","End":"05:02.875","Text":"It\u0027s intended to be odd numbers,"},{"Start":"05:02.875 ","End":"05:07.039","Text":"5, 7, 9, 11, 13,"},{"Start":"05:07.039 ","End":"05:11.000","Text":"and so on up to 27, and then squared."},{"Start":"05:11.000 ","End":"05:17.300","Text":"Now odd numbers can be written as 2n minus 1 or 2n plus 1."},{"Start":"05:17.300 ","End":"05:20.450","Text":"Let\u0027s take 2n minus 1 and it\u0027s squared."},{"Start":"05:20.450 ","End":"05:21.650","Text":"Now the question is,"},{"Start":"05:21.650 ","End":"05:23.870","Text":"what does n go from 2?"},{"Start":"05:23.870 ","End":"05:26.345","Text":"What are the upper and lower bounds of summation?"},{"Start":"05:26.345 ","End":"05:28.085","Text":"We want to get 5."},{"Start":"05:28.085 ","End":"05:29.930","Text":"If 2n minus 1 is 5,"},{"Start":"05:29.930 ","End":"05:31.940","Text":"then n is 3."},{"Start":"05:31.940 ","End":"05:37.685","Text":"We start at 3, then we get twice 3 minus 1 is 5 and how do we get the 27?"},{"Start":"05:37.685 ","End":"05:40.025","Text":"If you think about it, it\u0027s 14,"},{"Start":"05:40.025 ","End":"05:42.505","Text":"could add 1 and then divide by 2."},{"Start":"05:42.505 ","End":"05:43.875","Text":"Yeah, 14,"},{"Start":"05:43.875 ","End":"05:46.035","Text":"2n minus 1 is 27."},{"Start":"05:46.035 ","End":"05:47.955","Text":"That\u0027s the answer to this one."},{"Start":"05:47.955 ","End":"05:49.305","Text":"Now, the next one."},{"Start":"05:49.305 ","End":"05:50.990","Text":"If you just look at the denominators,"},{"Start":"05:50.990 ","End":"05:52.310","Text":"1 times 2, 2 times 3,"},{"Start":"05:52.310 ","End":"05:54.380","Text":"3 times 4, seems familiar."},{"Start":"05:54.380 ","End":"05:56.080","Text":"Let\u0027s go back."},{"Start":"05:56.080 ","End":"05:59.090","Text":"Yeah, we had something like that."},{"Start":"05:59.090 ","End":"06:01.820","Text":"1 times 2, 2 times 3, 3 times 4,"},{"Start":"06:01.820 ","End":"06:03.260","Text":"it ends in a different place,"},{"Start":"06:03.260 ","End":"06:07.160","Text":"but this part is the k, k plus one."},{"Start":"06:07.160 ","End":"06:09.504","Text":"Let\u0027s use a different letter."},{"Start":"06:09.504 ","End":"06:12.450","Text":"You can make it i, i plus 1,"},{"Start":"06:12.450 ","End":"06:14.220","Text":"and because it\u0027s on the denominator,"},{"Start":"06:14.220 ","End":"06:17.404","Text":"1 over and we just have to fill out the limits."},{"Start":"06:17.404 ","End":"06:23.070","Text":"We see it goes from 1 up to 10 so i goes from 1-10,"},{"Start":"06:23.070 ","End":"06:25.115","Text":"1 over i, i plus 1."},{"Start":"06:25.115 ","End":"06:27.020","Text":"I vary the index,"},{"Start":"06:27.020 ","End":"06:30.185","Text":"don\u0027t want to always use k. Because that\u0027s that 1."},{"Start":"06:30.185 ","End":"06:31.970","Text":"Now the next one,"},{"Start":"06:31.970 ","End":"06:33.785","Text":"you need to think about it a bit,"},{"Start":"06:33.785 ","End":"06:36.335","Text":"stare at it and see if you can find a pattern."},{"Start":"06:36.335 ","End":"06:40.355","Text":"The denominators look like they\u0027re powers of 3,"},{"Start":"06:40.355 ","End":"06:42.110","Text":"3 to the 1, 3 to the 2,"},{"Start":"06:42.110 ","End":"06:44.795","Text":"3 to the 3, and so on."},{"Start":"06:44.795 ","End":"06:47.245","Text":"What about the numerators?"},{"Start":"06:47.245 ","End":"06:55.160","Text":"They\u0027re gaps of 4 so what we could do is take multiples of 4 and adjust them."},{"Start":"06:55.160 ","End":"06:58.085","Text":"I could compare this to 4, 8,"},{"Start":"06:58.085 ","End":"07:02.230","Text":"12, 16, and then subtract 2."},{"Start":"07:02.230 ","End":"07:05.400","Text":"What I\u0027m suggesting is 4 minus 2,"},{"Start":"07:05.400 ","End":"07:07.770","Text":"8 minus 2, 12 minus 2,"},{"Start":"07:07.770 ","End":"07:10.230","Text":"up to 20 minus 2."},{"Start":"07:10.230 ","End":"07:13.380","Text":"Then the 4, 8, 12 are multiples of 4,"},{"Start":"07:13.380 ","End":"07:16.920","Text":"so we can write those as 4k and the denominators,"},{"Start":"07:16.920 ","End":"07:18.825","Text":"3 to the 1, 3 to the 2,"},{"Start":"07:18.825 ","End":"07:22.620","Text":"is 3 to the power of k. Just to make it clear,"},{"Start":"07:22.620 ","End":"07:26.180","Text":"we write the numerator as 4 times 1 minus 2, 4 times 2,"},{"Start":"07:26.180 ","End":"07:28.220","Text":"4 times 3, 4 times 4, 4 times 5,"},{"Start":"07:28.220 ","End":"07:30.740","Text":"4 times k minus 2,"},{"Start":"07:30.740 ","End":"07:32.975","Text":"and here 3 to the k. See the number is the same,"},{"Start":"07:32.975 ","End":"07:35.000","Text":"no more need for adjustment."},{"Start":"07:35.000 ","End":"07:38.700","Text":"4k minus 2 over 3 to the k,"},{"Start":"07:38.700 ","End":"07:41.160","Text":"and k goes from 1-5,"},{"Start":"07:41.160 ","End":"07:43.710","Text":"from 1, 2, 3, 4, to 5."},{"Start":"07:43.710 ","End":"07:45.360","Text":"Now the last one,"},{"Start":"07:45.360 ","End":"07:48.950","Text":"first one is fraction if you think of it as 4 over 1."},{"Start":"07:48.950 ","End":"07:51.290","Text":"On the numerators we have 4, 8, 12,"},{"Start":"07:51.290 ","End":"07:54.635","Text":"16, 20, looks like multiples of 4."},{"Start":"07:54.635 ","End":"07:57.180","Text":"On the denominator we\u0027d have 1,"},{"Start":"07:57.180 ","End":"08:01.560","Text":"5, 25, 125, so that\u0027s powers of 5,"},{"Start":"08:01.560 ","End":"08:04.155","Text":"starting with 5 to the 0."},{"Start":"08:04.155 ","End":"08:06.060","Text":"We can write it, denominators,"},{"Start":"08:06.060 ","End":"08:07.980","Text":"5 to the 0, 5 to the 1,"},{"Start":"08:07.980 ","End":"08:11.280","Text":"up to 5 to the 4th and here 4 times 1,"},{"Start":"08:11.280 ","End":"08:13.515","Text":"4 times 2, 4 times 3."},{"Start":"08:13.515 ","End":"08:17.510","Text":"We use it at a k and the question is whether we should let k be 1,"},{"Start":"08:17.510 ","End":"08:20.405","Text":"2, 3, or let it be 0, 1, 2."},{"Start":"08:20.405 ","End":"08:21.830","Text":"You could do it either way."},{"Start":"08:21.830 ","End":"08:24.200","Text":"If we let k go 0, 1, 2,"},{"Start":"08:24.200 ","End":"08:27.500","Text":"3, 4, then we\u0027ve got 5 to the k,"},{"Start":"08:27.500 ","End":"08:32.155","Text":"and k goes from 0 to 4 and then here it\u0027s k plus 1,"},{"Start":"08:32.155 ","End":"08:35.590","Text":"0 plus 1, 1 plus 1, 2 plus 1."},{"Start":"08:35.590 ","End":"08:37.550","Text":"But if we did it the other way,"},{"Start":"08:37.550 ","End":"08:43.940","Text":"we\u0027d get k goes from 1 to 5 but then the exponent would be k minus 1,"},{"Start":"08:43.940 ","End":"08:45.260","Text":"because if k is 1,"},{"Start":"08:45.260 ","End":"08:46.670","Text":"this is k minus 1,"},{"Start":"08:46.670 ","End":"08:48.815","Text":"k is 2, this is k minus 1."},{"Start":"08:48.815 ","End":"08:53.820","Text":"Either of these, and that\u0027s the last one and we\u0027re done."}],"ID":26640},{"Watched":false,"Name":"Exercise 3","Duration":"4m 34s","ChapterTopicVideoID":25837,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this exercise, there is 6 parts,"},{"Start":"00:02.565 ","End":"00:04.920","Text":"and each part we have to compute the sum."},{"Start":"00:04.920 ","End":"00:07.375","Text":"Let\u0027s start with Part a,"},{"Start":"00:07.375 ","End":"00:10.695","Text":"which is this and we\u0027ll need the formula,"},{"Start":"00:10.695 ","End":"00:12.870","Text":"this is the formula we\u0027re going to use."},{"Start":"00:12.870 ","End":"00:16.320","Text":"Now the first thing we\u0027ll do is use the rule for taking out a"},{"Start":"00:16.320 ","End":"00:20.055","Text":"constant so we can bring the 4 outside the Sigma."},{"Start":"00:20.055 ","End":"00:24.300","Text":"We have 4 times the sum k goes from 1 to 10 of k. For this,"},{"Start":"00:24.300 ","End":"00:28.305","Text":"we can use this formula where n is equal to 10."},{"Start":"00:28.305 ","End":"00:31.020","Text":"We have the 4 from here and then from this formula,"},{"Start":"00:31.020 ","End":"00:33.555","Text":"10 plus 1 over 2."},{"Start":"00:33.555 ","End":"00:36.080","Text":"The answer comes out to be 220."},{"Start":"00:36.080 ","End":"00:37.384","Text":"That\u0027s less important."},{"Start":"00:37.384 ","End":"00:40.870","Text":"Next part, we have the sum also from 1 to 10,"},{"Start":"00:40.870 ","End":"00:43.600","Text":"2k plus 4k squared."},{"Start":"00:43.600 ","End":"00:46.910","Text":"The formulas we\u0027ll need are this and this."},{"Start":"00:46.910 ","End":"00:49.670","Text":"We can break it up into the sum of"},{"Start":"00:49.670 ","End":"00:53.330","Text":"2 Sigmas and then bring the constants out in front we got."},{"Start":"00:53.330 ","End":"01:00.715","Text":"Then we\u0027ll use this formula with n equals 10 and this formula with n equals 10."},{"Start":"01:00.715 ","End":"01:03.390","Text":"Plugging in, this is what we get,"},{"Start":"01:03.390 ","End":"01:05.460","Text":"and this comes out to be 55,"},{"Start":"01:05.460 ","End":"01:10.700","Text":"this is 380 and all together it comes out to be 1,650 but as I said,"},{"Start":"01:10.700 ","End":"01:13.480","Text":"the actual answer is less important."},{"Start":"01:13.480 ","End":"01:15.510","Text":"Now the next part,"},{"Start":"01:15.510 ","End":"01:19.500","Text":"some from 10-24 of k, k minus 1."},{"Start":"01:19.500 ","End":"01:22.110","Text":"Notice it\u0027s not from 1-24."},{"Start":"01:22.110 ","End":"01:24.530","Text":"Formulas we\u0027ll need are here."},{"Start":"01:24.530 ","End":"01:28.520","Text":"What we\u0027ll do is expand the summand,"},{"Start":"01:28.520 ","End":"01:31.280","Text":"then we can break it up into 2 pieces."},{"Start":"01:31.280 ","End":"01:38.280","Text":"Got the sum of k squared minus the sum of k. Now we\u0027ll use the rule for splitting and"},{"Start":"01:38.280 ","End":"01:45.450","Text":"this will be from 1-24 minus the sum from 1-9 because we only want from 10 on wards,"},{"Start":"01:45.450 ","End":"01:46.935","Text":"so we subtract this part."},{"Start":"01:46.935 ","End":"01:49.820","Text":"Similarly here, but it\u0027s a minus,"},{"Start":"01:49.820 ","End":"01:51.575","Text":"so it\u0027s minus plus."},{"Start":"01:51.575 ","End":"01:55.285","Text":"Again from 1-24 and here from 1-9."},{"Start":"01:55.285 ","End":"02:01.115","Text":"The first 2 we can use this formula and for the last 2 we\u0027ll use this formula."},{"Start":"02:01.115 ","End":"02:03.080","Text":"Ones with n equals 24,"},{"Start":"02:03.080 ","End":"02:06.230","Text":"ones with n equals 9 and similarly for the other formula."},{"Start":"02:06.230 ","End":"02:14.420","Text":"I\u0027ll leave you to check the calculations and we\u0027ll go on to the next part, which is this."},{"Start":"02:14.420 ","End":"02:18.815","Text":"But I\u0027ve left the result of the previous part because we\u0027ll need it here."},{"Start":"02:18.815 ","End":"02:21.750","Text":"Now this will do some algebra first."},{"Start":"02:21.750 ","End":"02:26.255","Text":"The numerator, the k cubed minus k we can expand it."},{"Start":"02:26.255 ","End":"02:28.250","Text":"First of all we can take k out,"},{"Start":"02:28.250 ","End":"02:31.600","Text":"and then we can split this as a difference of squares,"},{"Start":"02:31.600 ","End":"02:33.185","Text":"so we have this."},{"Start":"02:33.185 ","End":"02:36.140","Text":"Then we can rewrite the numerator here as k,"},{"Start":"02:36.140 ","End":"02:41.345","Text":"k minus 1 k plus 1 and then something cancels the k plus 1."},{"Start":"02:41.345 ","End":"02:42.870","Text":"We have the sum k,"},{"Start":"02:42.870 ","End":"02:46.430","Text":"k minus 1 and this was exactly what we had in"},{"Start":"02:46.430 ","End":"02:51.175","Text":"the previous exercise so we can straight away write the answer."},{"Start":"02:51.175 ","End":"02:55.570","Text":"Next, this 1 and the formulas we\u0027re going to need are here."},{"Start":"02:55.570 ","End":"02:59.810","Text":"We\u0027ll expand k minus 2k plus 2 as a difference of squares,"},{"Start":"02:59.810 ","End":"03:01.610","Text":"k squared minus 4."},{"Start":"03:01.610 ","End":"03:05.480","Text":"We can separate this sum into 2 sums."},{"Start":"03:05.480 ","End":"03:09.530","Text":"Well, a difference, k squared and then the sum of 4."},{"Start":"03:09.530 ","End":"03:12.860","Text":"Now a sum from 4-10 is like from 1-10,"},{"Start":"03:12.860 ","End":"03:16.735","Text":"take away from 1-3 in both of these."},{"Start":"03:16.735 ","End":"03:19.980","Text":"Get from 1-10 of k squared minus 1-3."},{"Start":"03:19.980 ","End":"03:21.360","Text":"Then because of the minus we have here"},{"Start":"03:21.360 ","End":"03:27.285","Text":"a minus plus and then we use these formulas and we get,"},{"Start":"03:27.285 ","End":"03:31.460","Text":"for this and this, we use this formula with n equals 10 and n equals 3 and here"},{"Start":"03:31.460 ","End":"03:36.660","Text":"we have this formula twice 4 times 10 plus 4 times 3."},{"Start":"03:37.370 ","End":"03:42.495","Text":"I\u0027ll leave you to check the computation\u0027s, comes out 343."},{"Start":"03:42.495 ","End":"03:44.790","Text":"We have 1 more to go."},{"Start":"03:44.790 ","End":"03:46.440","Text":"That\u0027s this 1, the formulas,"},{"Start":"03:46.440 ","End":"03:48.810","Text":"we\u0027ll need 4 formulas."},{"Start":"03:48.810 ","End":"03:56.710","Text":"What will do is expand the brackets using algebra and this is the result."},{"Start":"03:56.710 ","End":"04:00.090","Text":"Then we can break up the sum into"},{"Start":"04:00.090 ","End":"04:04.715","Text":"4 separate sums and then we can take the constants out in front,"},{"Start":"04:04.715 ","End":"04:06.620","Text":"the 2 out, put the 4 out."},{"Start":"04:06.620 ","End":"04:09.200","Text":"Then we\u0027ll use each of these formulas."},{"Start":"04:09.200 ","End":"04:13.490","Text":"For this formula, we have this with n equals 10."},{"Start":"04:13.490 ","End":"04:18.020","Text":"For this, we have this formula with n equals 10,"},{"Start":"04:18.020 ","End":"04:21.335","Text":"then this formula with n equals 10,"},{"Start":"04:21.335 ","End":"04:24.030","Text":"and finally this formula."},{"Start":"04:24.520 ","End":"04:27.500","Text":"I\u0027ll leave you to check the arithmetic."},{"Start":"04:27.500 ","End":"04:30.380","Text":"Comes out to be 4,545,"},{"Start":"04:30.380 ","End":"04:35.190","Text":"and that\u0027s the last 1 in this set and we\u0027re done."}],"ID":26641},{"Watched":false,"Name":"Exercise 4","Duration":"3m 55s","ChapterTopicVideoID":25838,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have 3 computations to make."},{"Start":"00:03.570 ","End":"00:07.395","Text":"Each involves a summation with the Sigma notation."},{"Start":"00:07.395 ","End":"00:09.450","Text":"Start with part A."},{"Start":"00:09.450 ","End":"00:15.405","Text":"These are the formulas we\u0027ll be needing and just copy the exercise."},{"Start":"00:15.405 ","End":"00:17.430","Text":"Now, what we can do here,"},{"Start":"00:17.430 ","End":"00:19.305","Text":"just split the fraction,"},{"Start":"00:19.305 ","End":"00:21.570","Text":"make 2 separate fractions."},{"Start":"00:21.570 ","End":"00:23.775","Text":"Now, we can use this formula,"},{"Start":"00:23.775 ","End":"00:26.310","Text":"a^n over b^n is equal to a over b to the n."},{"Start":"00:26.310 ","End":"00:29.055","Text":"Here we can take 4 over 2,"},{"Start":"00:29.055 ","End":"00:34.305","Text":"which is 2^k and here 8 over 2, which is 4^k."},{"Start":"00:34.305 ","End":"00:35.930","Text":"This is what we get."},{"Start":"00:35.930 ","End":"00:38.030","Text":"Now we can use what we call linearity,"},{"Start":"00:38.030 ","End":"00:41.600","Text":"which means that Sigma take sums to sums."},{"Start":"00:41.600 ","End":"00:42.680","Text":"We can break it up as a sum"},{"Start":"00:42.680 ","End":"00:45.565","Text":"and you can also take constant in front of the Sigma."},{"Start":"00:45.565 ","End":"00:50.550","Text":"We have 5 times the sum of 2^k plus the sum of 4^k."},{"Start":"00:50.550 ","End":"00:53.135","Text":"Now, what we have is an instance of this,"},{"Start":"00:53.135 ","End":"00:58.105","Text":"we use this formula for the sum of from 1 to n of a^k."},{"Start":"00:58.105 ","End":"01:01.530","Text":"Here a will be 2 and k is 20,"},{"Start":"01:01.530 ","End":"01:03.690","Text":"here a is 4, k is 20."},{"Start":"01:03.690 ","End":"01:07.370","Text":"Just plugging in, we get this and could just do"},{"Start":"01:07.370 ","End":"01:11.320","Text":"this on the calculator or simplify a bit 2 minus 1 is 1,"},{"Start":"01:11.320 ","End":"01:13.530","Text":"and 5 times 2 is 10."},{"Start":"01:13.530 ","End":"01:17.835","Text":"Here we have 10 times 2^20 minus 1 and here,"},{"Start":"01:17.835 ","End":"01:19.290","Text":"4 minus 1 is 3,"},{"Start":"01:19.290 ","End":"01:21.225","Text":"so 4 over 3 times this,"},{"Start":"01:21.225 ","End":"01:23.215","Text":"and the rest of it will leave for the calculator."},{"Start":"01:23.215 ","End":"01:24.970","Text":"That\u0027s less important."},{"Start":"01:24.970 ","End":"01:28.905","Text":"Now on to part B, which is this."},{"Start":"01:28.905 ","End":"01:30.990","Text":"These are the formulas we\u0027ll need."},{"Start":"01:30.990 ","End":"01:35.340","Text":"What I\u0027ll do first is I don\u0027t like this k plus 2 here,"},{"Start":"01:35.340 ","End":"01:39.750","Text":"it\u0027ll be nice if it was just 4^k because then we could use this formula."},{"Start":"01:39.750 ","End":"01:41.970","Text":"We can break this up,"},{"Start":"01:41.970 ","End":"01:45.405","Text":"4^k plus 2 is 4 squared 4^k."},{"Start":"01:45.405 ","End":"01:49.835","Text":"Also broke the fraction up this over this plus this over this."},{"Start":"01:49.835 ","End":"01:52.355","Text":"Now, what we can do is use this formula,"},{"Start":"01:52.355 ","End":"02:00.580","Text":"4 over 0.4^k and then 10 over 0.4^k."},{"Start":"02:00.580 ","End":"02:03.225","Text":"The 2 times 4 squared is 32."},{"Start":"02:03.225 ","End":"02:06.074","Text":"4 Over 0.4 is 10,"},{"Start":"02:06.074 ","End":"02:11.145","Text":"and 10 over 0.4 is 25. We\u0027ve got this."},{"Start":"02:11.145 ","End":"02:14.300","Text":"Now, we\u0027ll split this up into the sum and also"},{"Start":"02:14.300 ","End":"02:17.555","Text":"take the constant in front of the Sigma. We have this."},{"Start":"02:17.555 ","End":"02:19.880","Text":"Now, we want to use this formula twice,"},{"Start":"02:19.880 ","End":"02:24.660","Text":"once with a equals 10 and once with a equals 25, just substituting."},{"Start":"02:24.660 ","End":"02:27.315","Text":"This is the expression we get."},{"Start":"02:27.315 ","End":"02:29.210","Text":"Just leave it there."},{"Start":"02:29.210 ","End":"02:31.099","Text":"We can just do the rest by calculator."},{"Start":"02:31.099 ","End":"02:33.710","Text":"This is not the point of the exercise is to work with"},{"Start":"02:33.710 ","End":"02:36.620","Text":"Sigma\u0027s rather than to get the actual numerical answer."},{"Start":"02:36.620 ","End":"02:38.675","Text":"Let\u0027s go on to part C,"},{"Start":"02:38.675 ","End":"02:40.400","Text":"which is this,"},{"Start":"02:40.400 ","End":"02:43.235","Text":"and here\u0027s the formulas we\u0027re going to need,"},{"Start":"02:43.235 ","End":"02:46.610","Text":"I\u0027d like to have something just to the power of k."},{"Start":"02:46.610 ","End":"02:51.568","Text":"First thing we can do is split it up into a product"},{"Start":"02:51.568 ","End":"02:55.445","Text":"using laws of exponents 2^2k times 2^10."},{"Start":"02:55.445 ","End":"03:00.725","Text":"Now we can say this is 2 squared to the power of k using this formula."},{"Start":"03:00.725 ","End":"03:04.250","Text":"This will be 4^k, 4 is 2 squared."},{"Start":"03:04.250 ","End":"03:07.250","Text":"Then 2^10 is a constant it doesn\u0027t have k in it,"},{"Start":"03:07.250 ","End":"03:09.185","Text":"so we can pull that in front."},{"Start":"03:09.185 ","End":"03:11.570","Text":"We\u0027re not quite ready to use this formula,"},{"Start":"03:11.570 ","End":"03:14.180","Text":"because this formula for when k goes from 1 to n,"},{"Start":"03:14.180 ","End":"03:18.873","Text":"here we have 10-20, so we can use the splitting rule"},{"Start":"03:18.873 ","End":"03:23.970","Text":"and make it from 1-20 minus the sum from 1-9."},{"Start":"03:23.970 ","End":"03:25.730","Text":"Then we can use this rule twice,"},{"Start":"03:25.730 ","End":"03:27.970","Text":"once with n equals 20,"},{"Start":"03:27.970 ","End":"03:30.335","Text":"once with n equals 9,"},{"Start":"03:30.335 ","End":"03:32.615","Text":"and this is what we get."},{"Start":"03:32.615 ","End":"03:33.995","Text":"Now simplify a bit."},{"Start":"03:33.995 ","End":"03:38.210","Text":"The denominators here are both 3 in each case we have 4 over 3."},{"Start":"03:38.210 ","End":"03:40.375","Text":"We can bring 4 over 3 out,"},{"Start":"03:40.375 ","End":"03:42.785","Text":"and then 4^20 minus 1,"},{"Start":"03:42.785 ","End":"03:44.880","Text":"takeaway, 4^9 minus 1,"},{"Start":"03:44.880 ","End":"03:46.455","Text":"the minus 1 cancels."},{"Start":"03:46.455 ","End":"03:49.140","Text":"It\u0027s just 4^20 minus 4^9."},{"Start":"03:49.140 ","End":"03:50.480","Text":"At this point I\u0027ll say,"},{"Start":"03:50.480 ","End":"03:55.170","Text":"it\u0027s a job for the calculator and we\u0027ll just stop here."}],"ID":26642},{"Watched":false,"Name":"Exercise 5","Duration":"3m 5s","ChapterTopicVideoID":25839,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"This exercise has 4 parts,"},{"Start":"00:02.940 ","End":"00:06.570","Text":"and each of them we have a sum with a dot,"},{"Start":"00:06.570 ","End":"00:08.504","Text":"dot, dot, an ellipses."},{"Start":"00:08.504 ","End":"00:10.410","Text":"We have to compute each of these."},{"Start":"00:10.410 ","End":"00:12.855","Text":"I want the actual numerical result,"},{"Start":"00:12.855 ","End":"00:15.270","Text":"and we\u0027ll do them by writing them in the form of"},{"Start":"00:15.270 ","End":"00:18.945","Text":"Sigma and then using the standard formulas."},{"Start":"00:18.945 ","End":"00:24.600","Text":"In part a, we can write this as the sum of k squared,"},{"Start":"00:24.600 ","End":"00:27.135","Text":"where k goes from 1-20,"},{"Start":"00:27.135 ","End":"00:33.675","Text":"and this is the formula we can use with n equals 20."},{"Start":"00:33.675 ","End":"00:35.700","Text":"Just plugging it in."},{"Start":"00:35.700 ","End":"00:42.400","Text":"This is the expression we get and the result comes out to be 2870."},{"Start":"00:42.400 ","End":"00:45.800","Text":"Now part b, which is this,"},{"Start":"00:45.800 ","End":"00:50.555","Text":"we can\u0027t use this formula right away because this would be,"},{"Start":"00:50.555 ","End":"00:52.310","Text":"if we started from 1,"},{"Start":"00:52.310 ","End":"00:56.015","Text":"we have the sum from 4 to 24 here."},{"Start":"00:56.015 ","End":"01:02.720","Text":"But what we can do is break it up from 1-24 minus from 1-3."},{"Start":"01:02.720 ","End":"01:06.935","Text":"Then we can apply this formula 1\u0027s with 24 and 1\u0027s with 3,"},{"Start":"01:06.935 ","End":"01:09.650","Text":"get this minus this,"},{"Start":"01:09.650 ","End":"01:11.750","Text":"which comes out to this,"},{"Start":"01:11.750 ","End":"01:15.050","Text":"you\u0027re going to be using the calculator or pencil and paper,"},{"Start":"01:15.050 ","End":"01:19.015","Text":"and the answer comes out to be 4,886."},{"Start":"01:19.015 ","End":"01:21.140","Text":"Now, part c here,"},{"Start":"01:21.140 ","End":"01:22.190","Text":"it\u0027s a bit different."},{"Start":"01:22.190 ","End":"01:23.890","Text":"We only have even numbers."},{"Start":"01:23.890 ","End":"01:25.040","Text":"That seems to be the pattern."},{"Start":"01:25.040 ","End":"01:28.285","Text":"Even numbers from 2 up to 22 squared."},{"Start":"01:28.285 ","End":"01:29.540","Text":"We\u0027ll need this formula,"},{"Start":"01:29.540 ","End":"01:31.340","Text":"but we can\u0027t use it right away."},{"Start":"01:31.340 ","End":"01:33.150","Text":"What we can do is write the even numbers as"},{"Start":"01:33.150 ","End":"01:36.695","Text":"2 times something can we have from 1 up to 11."},{"Start":"01:36.695 ","End":"01:41.450","Text":"This is the sum of 2k squared where k goes from 1,"},{"Start":"01:41.450 ","End":"01:43.384","Text":"2, 3 up to 11."},{"Start":"01:43.384 ","End":"01:47.730","Text":"2k squared is 4k squared and we can bring the 4 out in front."},{"Start":"01:47.730 ","End":"01:50.959","Text":"Now, we can apply this formula with n equals 11,"},{"Start":"01:50.959 ","End":"01:52.430","Text":"and we get this,"},{"Start":"01:52.430 ","End":"01:56.845","Text":"and then computation gives us 2,024."},{"Start":"01:56.845 ","End":"01:59.535","Text":"Part d, this is it."},{"Start":"01:59.535 ","End":"02:02.345","Text":"We\u0027ll need 3 formulas."},{"Start":"02:02.345 ","End":"02:08.120","Text":"What we\u0027ll do is write each odd number as 2k minus 1."},{"Start":"02:08.120 ","End":"02:10.010","Text":"1 is twice 1 minus 1,"},{"Start":"02:10.010 ","End":"02:11.735","Text":"twice 2 minus 1."},{"Start":"02:11.735 ","End":"02:14.180","Text":"It\u0027s 2k minus 1 squared."},{"Start":"02:14.180 ","End":"02:16.760","Text":"There\u0027s a general pattern and k goes from 1,"},{"Start":"02:16.760 ","End":"02:19.080","Text":"2, 3 up to 9."},{"Start":"02:19.080 ","End":"02:23.000","Text":"Let\u0027s expand this as a quadratic expression."},{"Start":"02:23.000 ","End":"02:25.760","Text":"It\u0027s 4k squared minus 4k plus 1,"},{"Start":"02:25.760 ","End":"02:27.170","Text":"and then break it up,"},{"Start":"02:27.170 ","End":"02:32.540","Text":"4 times the sum of k squared minus 4 times the sum of k plus the sum of 1."},{"Start":"02:32.540 ","End":"02:35.870","Text":"We\u0027ll need all these 3 formulas."},{"Start":"02:35.870 ","End":"02:40.100","Text":"This formula with n equals 9 gives us this."},{"Start":"02:40.100 ","End":"02:44.590","Text":"This formula with n equals 9 gives us this."},{"Start":"02:44.590 ","End":"02:48.665","Text":"Of course, we have the minus 4 also and the 4 in front here and here."},{"Start":"02:48.665 ","End":"02:53.770","Text":"We have this formula with c equals 1 and equals 9. We got this."},{"Start":"02:53.770 ","End":"02:58.370","Text":"Altogether, use the calculator to do each of these,"},{"Start":"02:58.370 ","End":"03:01.310","Text":"the final answer is 969,"},{"Start":"03:01.310 ","End":"03:05.940","Text":"and that\u0027s the last part of the 4th and that concludes this clip."}],"ID":26643},{"Watched":false,"Name":"Exercise 6","Duration":"2m 40s","ChapterTopicVideoID":25840,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.745","Text":"This exercise, there are 2 parts,"},{"Start":"00:02.745 ","End":"00:09.025","Text":"and each part is a proof and both of them will be done using an index shift,"},{"Start":"00:09.025 ","End":"00:10.910","Text":"and we have formulas for that."},{"Start":"00:10.910 ","End":"00:12.655","Text":"I won\u0027t read them out."},{"Start":"00:12.655 ","End":"00:15.580","Text":"Let\u0027s just start with part a,"},{"Start":"00:15.580 ","End":"00:17.230","Text":"just copied it here."},{"Start":"00:17.230 ","End":"00:19.105","Text":"We have to show that this equals this."},{"Start":"00:19.105 ","End":"00:22.465","Text":"At the moment, I put a question mark, is it equal?"},{"Start":"00:22.465 ","End":"00:26.065","Text":"What we\u0027ll do is an index shift,"},{"Start":"00:26.065 ","End":"00:29.080","Text":"since this is k from 1 to n. Here,"},{"Start":"00:29.080 ","End":"00:30.815","Text":"k starts from 0,"},{"Start":"00:30.815 ","End":"00:35.860","Text":"the obvious thing to do is to decrease the bounds by 1."},{"Start":"00:35.860 ","End":"00:38.140","Text":"If we decrease the bounds by 1,"},{"Start":"00:38.140 ","End":"00:42.550","Text":"you then replace k by k plus 1 in the summand."},{"Start":"00:42.550 ","End":"00:48.290","Text":"We\u0027ll start with the left-hand side and what we\u0027ll do is subtract 1 from the bound."},{"Start":"00:48.290 ","End":"00:51.995","Text":"Here, we have 1 minus 1 and n minus 1,"},{"Start":"00:51.995 ","End":"00:57.840","Text":"and to compensate, we replace k by k plus 1."},{"Start":"00:58.120 ","End":"01:01.565","Text":"Just do the subtraction of 1,"},{"Start":"01:01.565 ","End":"01:05.110","Text":"and we have k goes from 0 to n minus 1,"},{"Start":"01:05.110 ","End":"01:06.860","Text":"and then just tidy up here;"},{"Start":"01:06.860 ","End":"01:08.490","Text":"k plus 1 plus 2 is k plus 3."},{"Start":"01:08.490 ","End":"01:13.595","Text":"Similarly here, we have 2k plus 2 plus 4 or 2k plus 6."},{"Start":"01:13.595 ","End":"01:15.365","Text":"When you look at this,"},{"Start":"01:15.365 ","End":"01:18.575","Text":"this is exactly what\u0027s written on the right-hand side,"},{"Start":"01:18.575 ","End":"01:21.485","Text":"so that concludes part a."},{"Start":"01:21.485 ","End":"01:28.595","Text":"Part b, I want to be pedantic when I add a condition that n is bigger or equal to 7."},{"Start":"01:28.595 ","End":"01:32.960","Text":"The upper bound has to be bigger or equal to the lower bound,"},{"Start":"01:32.960 ","End":"01:36.070","Text":"so n minus 3 has to be bigger or equal to 4,"},{"Start":"01:36.070 ","End":"01:38.390","Text":"and that gives n bigger or equal to 7."},{"Start":"01:38.390 ","End":"01:40.695","Text":"I won\u0027t use this in the proof,"},{"Start":"01:40.695 ","End":"01:43.190","Text":"but to be technically precise,"},{"Start":"01:43.190 ","End":"01:45.110","Text":"we should state this."},{"Start":"01:45.110 ","End":"01:49.490","Text":"Part b, we have to prove this equals this."},{"Start":"01:49.490 ","End":"01:51.785","Text":"Again, we\u0027ll use an index shift,"},{"Start":"01:51.785 ","End":"01:57.005","Text":"but this time the upper and lower bounds are going to shift up by 4,"},{"Start":"01:57.005 ","End":"02:00.185","Text":"and then we\u0027re going to decrease the index by 4."},{"Start":"02:00.185 ","End":"02:06.480","Text":"We increase here and here by 4 and replace k by k minus 4."},{"Start":"02:06.480 ","End":"02:11.430","Text":"We have that this equals plus 4 here, plus 4 here."},{"Start":"02:11.430 ","End":"02:13.620","Text":"Here, we have k minus 4,"},{"Start":"02:13.620 ","End":"02:16.575","Text":"k minus 4, k minus 4."},{"Start":"02:16.575 ","End":"02:18.810","Text":"Now we just have to tidy up;"},{"Start":"02:18.810 ","End":"02:21.375","Text":"4 plus 4 is 8,"},{"Start":"02:21.375 ","End":"02:24.060","Text":"n minus 3 plus 4 is n plus 1,"},{"Start":"02:24.060 ","End":"02:28.155","Text":"4k minus 16 plus 17 is 4k plus 1."},{"Start":"02:28.155 ","End":"02:31.220","Text":"Here twice k minus 4 is 2k minus 8,"},{"Start":"02:31.220 ","End":"02:33.770","Text":"and k minus 4 plus 1 is k minus 3."},{"Start":"02:33.770 ","End":"02:37.265","Text":"After all this, we see we\u0027ve got exactly the right-hand side,"},{"Start":"02:37.265 ","End":"02:40.680","Text":"so that proves it and we are done."}],"ID":26644},{"Watched":false,"Name":"Exercise 7","Duration":"3m 16s","ChapterTopicVideoID":25841,"CourseChapterTopicPlaylistID":246310,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"This exercise, there\u0027s 2 parts."},{"Start":"00:03.105 ","End":"00:05.910","Text":"In each part we have to compute the sum."},{"Start":"00:05.910 ","End":"00:10.710","Text":"But by shifting the index rather than by splitting the sum."},{"Start":"00:10.710 ","End":"00:13.455","Text":"I\u0027ll explain as we get into it."},{"Start":"00:13.455 ","End":"00:17.310","Text":"In Part A, these are the formulas we need."},{"Start":"00:17.310 ","End":"00:21.015","Text":"I\u0027ll show you what I mean by letting the sum."},{"Start":"00:21.015 ","End":"00:22.815","Text":"That would be to say,"},{"Start":"00:22.815 ","End":"00:24.600","Text":"\"We need from 4-11,"},{"Start":"00:24.600 ","End":"00:29.520","Text":"so we\u0027ll take from 1-11 and take away the sum from 1-3.\""},{"Start":"00:29.520 ","End":"00:31.890","Text":"This is the way not to do it."},{"Start":"00:31.890 ","End":"00:34.410","Text":"Shifting the index means to make"},{"Start":"00:34.410 ","End":"00:39.435","Text":"a substitution so that it comes out to be something from 1 to something."},{"Start":"00:39.435 ","End":"00:42.270","Text":"Like here., we wanted to start with 1."},{"Start":"00:42.270 ","End":"00:45.035","Text":"We subtract 3 here and here,"},{"Start":"00:45.035 ","End":"00:50.645","Text":"and we compensate by replacing i with i plus 3 and what we get,"},{"Start":"00:50.645 ","End":"00:52.385","Text":"7 minus 3 is 8,"},{"Start":"00:52.385 ","End":"00:53.810","Text":"4 minus 3 is 1."},{"Start":"00:53.810 ","End":"00:58.640","Text":"So we have a 1-8 and the 9 plus 3 squared is a squared plus 6 i plus 9."},{"Start":"00:58.640 ","End":"01:02.165","Text":"Now we can split this up using linearity."},{"Start":"01:02.165 ","End":"01:08.180","Text":"We break it up into 3 sums and we also take the constants outside the sigma, like here."},{"Start":"01:08.180 ","End":"01:11.345","Text":"Then each of these has its own formula."},{"Start":"01:11.345 ","End":"01:13.970","Text":"For the first bit, the I squared,"},{"Start":"01:13.970 ","End":"01:17.120","Text":"we\u0027ll use this formula with n equals 8."},{"Start":"01:17.120 ","End":"01:18.930","Text":"So we get this."},{"Start":"01:18.930 ","End":"01:21.575","Text":"Here we have 6 times and then the sum,"},{"Start":"01:21.575 ","End":"01:24.555","Text":"this 1 with n equals 8."},{"Start":"01:24.555 ","End":"01:29.450","Text":"Then here we have the constant 9 times 8."},{"Start":"01:29.450 ","End":"01:31.900","Text":"Do this using the calculator."},{"Start":"01:31.900 ","End":"01:33.595","Text":"I did a little bit in my head."},{"Start":"01:33.595 ","End":"01:36.070","Text":"Here the actual answer is not important."},{"Start":"01:36.070 ","End":"01:38.155","Text":"Now Part B,"},{"Start":"01:38.155 ","End":"01:40.600","Text":"and this is the formula we\u0027ll need."},{"Start":"01:40.600 ","End":"01:44.890","Text":"I remind you we\u0027re going to do it not by splitting the sum."},{"Start":"01:44.890 ","End":"01:46.870","Text":"If we split the sum, this is how we would do it."},{"Start":"01:46.870 ","End":"01:53.475","Text":"Who would say that the sum from 10-20 is like from 1-20, takeaway from 1-9."},{"Start":"01:53.475 ","End":"01:55.045","Text":"That\u0027s not the way we\u0027re going to do it."},{"Start":"01:55.045 ","End":"01:56.770","Text":"We\u0027re going to do an index Shift,"},{"Start":"01:56.770 ","End":"02:00.685","Text":"we\u0027ll shift J so that it starts from 1 instead of from 10."},{"Start":"02:00.685 ","End":"02:03.910","Text":"So what we\u0027ll do is we\u0027ll subtract 9 from"},{"Start":"02:03.910 ","End":"02:10.790","Text":"the upper and lower bounds and compensate by replacing j with j plus 9."},{"Start":"02:10.790 ","End":"02:13.230","Text":"10 minus 9 is 1,"},{"Start":"02:13.230 ","End":"02:15.215","Text":"20, minus 9 is 11."},{"Start":"02:15.215 ","End":"02:21.080","Text":"What I also did was to apply this formula with a is 4 and m is 2."},{"Start":"02:21.080 ","End":"02:25.525","Text":"So 4 squared to the power of n is 4 to the 2n."},{"Start":"02:25.525 ","End":"02:28.080","Text":"So 4 squared is 16."},{"Start":"02:28.080 ","End":"02:29.530","Text":"So this gets rid of the 2."},{"Start":"02:29.530 ","End":"02:31.370","Text":"Now want to get rid of the plus 9."},{"Start":"02:31.370 ","End":"02:35.090","Text":"I want to say something to the power of a simple index."},{"Start":"02:35.090 ","End":"02:40.545","Text":"So what we can do is just use the rules of exponents 16 to the j, 16 to the 9."},{"Start":"02:40.545 ","End":"02:42.530","Text":"16 to the 9 is a constant."},{"Start":"02:42.530 ","End":"02:43.985","Text":"It doesn\u0027t contain j,"},{"Start":"02:43.985 ","End":"02:46.435","Text":"so we can bring it out in front."},{"Start":"02:46.435 ","End":"02:49.775","Text":"Now we can finally apply the formula,"},{"Start":"02:49.775 ","End":"02:55.265","Text":"the sum of a to the j and n is 11 here."},{"Start":"02:55.265 ","End":"02:57.710","Text":"So we get 16,"},{"Start":"02:57.710 ","End":"03:00.755","Text":"16 to the 11 minus 1, 16 minus 1,"},{"Start":"03:00.755 ","End":"03:07.820","Text":"16 to the 9 times 16 is 16 to the 10th and 16 minus 1 is 15."},{"Start":"03:07.820 ","End":"03:10.700","Text":"That\u0027s enough. Do it by calculator,"},{"Start":"03:10.700 ","End":"03:12.530","Text":"it doesn\u0027t really matter what the answer is."},{"Start":"03:12.530 ","End":"03:17.040","Text":"It\u0027s the method that\u0027s important here. Okay, so we\u0027re done."}],"ID":26645}],"Thumbnail":null,"ID":246310},{"Name":"Famous Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Arithmetic-Geometric Mean Inequality","Duration":"4m 31s","ChapterTopicVideoID":25893,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"A new topic, Inequalities"},{"Start":"00:02.700 ","End":"00:06.030","Text":"and we\u0027re going to talk about 4 famous ones."},{"Start":"00:06.030 ","End":"00:09.150","Text":"In this clip, start with the first of the 4,"},{"Start":"00:09.150 ","End":"00:13.605","Text":"which relates to the Arithmetic-Geometric Means."},{"Start":"00:13.605 ","End":"00:16.800","Text":"This is also called the AM-GM Inequality,"},{"Start":"00:16.800 ","End":"00:18.915","Text":"arithmetic mean, geometric mean."},{"Start":"00:18.915 ","End":"00:21.990","Text":"Let a, b be 2 real numbers,"},{"Start":"00:21.990 ","End":"00:29.220","Text":"preferably positive and we define the arithmetic mean as it\u0027s known as the average."},{"Start":"00:29.220 ","End":"00:31.905","Text":"You add them up and divide by 2."},{"Start":"00:31.905 ","End":"00:34.770","Text":"For example, if a is 3 and b is 6,"},{"Start":"00:34.770 ","End":"00:39.945","Text":"then the arithmetic mean is 3 plus 6/2 4.5."},{"Start":"00:39.945 ","End":"00:45.110","Text":"The geometric mean is defined as the square root of ab,"},{"Start":"00:45.110 ","End":"00:46.759","Text":"since a and b are positive,"},{"Start":"00:46.759 ","End":"00:50.185","Text":"so is ab and then it will have a square root."},{"Start":"00:50.185 ","End":"00:53.220","Text":"For example, a is 3,"},{"Start":"00:53.220 ","End":"00:54.585","Text":"b is 6,"},{"Start":"00:54.585 ","End":"01:00.740","Text":"so the geometric mean is the square root of 3 times 6, square root of 18,"},{"Start":"01:00.740 ","End":"01:09.860","Text":"roughly 4.24 unless known is the harmonic mean it\u0027s defined by this formula,"},{"Start":"01:09.860 ","End":"01:12.245","Text":"2 over 1/a plus 1/b,"},{"Start":"01:12.245 ","End":"01:17.320","Text":"which can be simplified to 2ab over a plus b."},{"Start":"01:17.320 ","End":"01:20.570","Text":"Let\u0027s take the same a and b as here and here,"},{"Start":"01:20.570 ","End":"01:22.250","Text":"a is 3, b is 6."},{"Start":"01:22.250 ","End":"01:25.100","Text":"The harmonic mean is,"},{"Start":"01:25.100 ","End":"01:26.240","Text":"using this formula,"},{"Start":"01:26.240 ","End":"01:30.740","Text":"2 times 3 times 6/3 plus 6 comes out to be 4."},{"Start":"01:30.740 ","End":"01:33.620","Text":"Term harmonic often relates to reciprocal"},{"Start":"01:33.620 ","End":"01:35.180","Text":"and in fact,"},{"Start":"01:35.180 ","End":"01:37.790","Text":"harmonic mean is like an arithmetic mean,"},{"Start":"01:37.790 ","End":"01:39.575","Text":"but with reciprocals."},{"Start":"01:39.575 ","End":"01:44.900","Text":"It turns out if c is the harmonic mean of a and b as the same thing as"},{"Start":"01:44.900 ","End":"01:50.495","Text":"saying 1 over c is the arithmetic mean of 1/a and 1/b."},{"Start":"01:50.495 ","End":"01:56.150","Text":"Given numerical example, we can use this to actually calculate the harmonic mean."},{"Start":"01:56.150 ","End":"01:58.505","Text":"We start with 2 numbers like 3 and 6."},{"Start":"01:58.505 ","End":"02:02.105","Text":"Take the reciprocals, 1/3 and 1/6."},{"Start":"02:02.105 ","End":"02:04.565","Text":"Then take the arithmetic mean."},{"Start":"02:04.565 ","End":"02:10.670","Text":"A 1/3 plus a 1/6 is a half divided by 2 is a 1/4 and then undo the reciprocal,"},{"Start":"02:10.670 ","End":"02:13.025","Text":"which is also reciprocal and we get to 4."},{"Start":"02:13.025 ","End":"02:15.745","Text":"From 3 and 6 we get to 4."},{"Start":"02:15.745 ","End":"02:17.954","Text":"Harmonic mean."},{"Start":"02:17.954 ","End":"02:22.715","Text":"I will give you an example of its use in time distance problems."},{"Start":"02:22.715 ","End":"02:28.310","Text":"If an object travels from point a to point b at an average speed of 3 meters per"},{"Start":"02:28.310 ","End":"02:35.000","Text":"second and returns from b to a at an average speed of 6 meters per second."},{"Start":"02:35.000 ","End":"02:38.090","Text":"Then the average speed on the round trip is"},{"Start":"02:38.090 ","End":"02:42.635","Text":"not the regular arithmetic mean. It\u0027s not 4.5."},{"Start":"02:42.635 ","End":"02:46.495","Text":"It\u0027s actually the harmonic mean 4."},{"Start":"02:46.495 ","End":"02:48.080","Text":"If you\u0027re not sure about this,"},{"Start":"02:48.080 ","End":"02:50.690","Text":"you should sit with pencil and paper and actually"},{"Start":"02:50.690 ","End":"02:54.170","Text":"solve this to see that we really do need the harmonic mean,"},{"Start":"02:54.170 ","End":"02:57.755","Text":"or that the answer is 4 meters per second."},{"Start":"02:57.755 ","End":"03:02.525","Text":"Anyway, here\u0027s the important proposition or claim."},{"Start":"03:02.525 ","End":"03:07.460","Text":"In general, the harmonic mean is less than or equal to the geometric mean,"},{"Start":"03:07.460 ","End":"03:10.205","Text":"which is less than or equal to the arithmetic mean,"},{"Start":"03:10.205 ","End":"03:13.505","Text":"not just in this particular example."},{"Start":"03:13.505 ","End":"03:18.275","Text":"Well, let\u0027s first all see that it is true here and it works out here,"},{"Start":"03:18.275 ","End":"03:23.210","Text":"but it\u0027s also true in general and when I say in general,"},{"Start":"03:23.210 ","End":"03:26.305","Text":"we can also generalize it to more than 2 numbers."},{"Start":"03:26.305 ","End":"03:29.400","Text":"Let\u0027s suppose we have n numbers, x_1,"},{"Start":"03:29.400 ","End":"03:32.540","Text":"x_2 up to x_n and it\u0027s easier if"},{"Start":"03:32.540 ","End":"03:35.810","Text":"they\u0027re positive real numbers that way when we take square root,"},{"Start":"03:35.810 ","End":"03:40.385","Text":"you won\u0027t have a problem, or if we take reciprocal won\u0027t be dividing by 0."},{"Start":"03:40.385 ","End":"03:43.365","Text":"Usually stick to positive real numbers."},{"Start":"03:43.365 ","End":"03:45.020","Text":"There are 3 means."},{"Start":"03:45.020 ","End":"03:49.630","Text":"The arithmetic mean, just a generalization of the formula with 2"},{"Start":"03:49.630 ","End":"03:54.200","Text":"which is we just add them all up and divide by n. The geometric mean,"},{"Start":"03:54.200 ","End":"03:56.960","Text":"multiply them all together, that is the square root."},{"Start":"03:56.960 ","End":"04:03.275","Text":"We take the nth root if there\u0027s n factors and the harmonic mean,"},{"Start":"04:03.275 ","End":"04:06.020","Text":"we add up the reciprocals,"},{"Start":"04:06.020 ","End":"04:08.960","Text":"divide by n and then take the reciprocal of that."},{"Start":"04:08.960 ","End":"04:10.415","Text":"So the n goes on top."},{"Start":"04:10.415 ","End":"04:14.840","Text":"Anyway, this is the formula for the harmonic mean and the proposition"},{"Start":"04:14.840 ","End":"04:19.565","Text":"should really be stated in general with n numbers."},{"Start":"04:19.565 ","End":"04:22.430","Text":"Harmonic means less than or equal to geometric mean,"},{"Start":"04:22.430 ","End":"04:28.455","Text":"less than or equal to the arithmetic mean and we\u0027ll prove this in the exercises."},{"Start":"04:28.455 ","End":"04:31.310","Text":"That\u0027s it for this clip."}],"ID":26703},{"Watched":false,"Name":"Triangle Inequality","Duration":"4m 46s","ChapterTopicVideoID":25892,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.790","Text":"Continuing with famous inequalities,"},{"Start":"00:02.790 ","End":"00:05.130","Text":"the next 1 is the triangle inequality,"},{"Start":"00:05.130 ","End":"00:09.340","Text":"which actually comes in several variants."},{"Start":"00:09.440 ","End":"00:14.100","Text":"The role related to the absolute value of the sum"},{"Start":"00:14.100 ","End":"00:16.255","Text":"or difference of 2 numbers."},{"Start":"00:16.255 ","End":"00:19.430","Text":"I want to remind you what the absolute value is,"},{"Start":"00:19.430 ","End":"00:23.330","Text":"we define the absolute value of a real number to be itself."},{"Start":"00:23.330 ","End":"00:25.720","Text":"If it\u0027s bigger or equal to 0,"},{"Start":"00:25.720 ","End":"00:29.120","Text":"and we negate it if it\u0027s negative."},{"Start":"00:29.120 ","End":"00:31.310","Text":"So this will come out to be positive."},{"Start":"00:31.310 ","End":"00:34.880","Text":"Some examples, absolute value of 4 is 4,"},{"Start":"00:34.880 ","End":"00:36.920","Text":"because 4 is bigger or equal to 0."},{"Start":"00:36.920 ","End":"00:39.365","Text":"Absolute value of 0 is 0,"},{"Start":"00:39.365 ","End":"00:41.555","Text":"because 0 is bigger or equal to 0."},{"Start":"00:41.555 ","End":"00:45.470","Text":"Then in the case of a negative number absolute value of minus 4,"},{"Start":"00:45.470 ","End":"00:46.490","Text":"according to this formula,"},{"Start":"00:46.490 ","End":"00:50.545","Text":"it\u0027s minus minus 4 in other words it\u0027s plus 4."},{"Start":"00:50.545 ","End":"00:55.385","Text":"Absolute value always gives us a positive number, or possibly 0."},{"Start":"00:55.385 ","End":"00:59.040","Text":"Here are some of the properties of the absolute value."},{"Start":"00:59.040 ","End":"01:03.700","Text":"The absolute value of minus a is the same as the absolute value of a."},{"Start":"01:03.700 ","End":"01:05.890","Text":"Here, for example, the absolute value of"},{"Start":"01:05.890 ","End":"01:08.620","Text":"4 and the absolute value of minus 4 are the same,"},{"Start":"01:08.620 ","End":"01:10.630","Text":"they both come out to be 4."},{"Start":"01:10.630 ","End":"01:16.430","Text":"The absolute value of a minus b is also the distance from a to b,"},{"Start":"01:16.430 ","End":"01:20.810","Text":"so the absolute value of 7 minus 3 would be 4."},{"Start":"01:20.810 ","End":"01:23.930","Text":"Also the absolute value of 3 minus 7 would be 4,"},{"Start":"01:23.930 ","End":"01:26.915","Text":"because the absolute value of minus 4, which is 4."},{"Start":"01:26.915 ","End":"01:29.935","Text":"Distance from a to b is the same as distance from b to a."},{"Start":"01:29.935 ","End":"01:31.720","Text":"It preserves products"},{"Start":"01:31.720 ","End":"01:35.229","Text":"because really we multiply and ignore the sign."},{"Start":"01:35.229 ","End":"01:38.055","Text":"Division is like multiplication,"},{"Start":"01:38.055 ","End":"01:41.655","Text":"but we have to watch out, of course not to divide by 0."},{"Start":"01:41.655 ","End":"01:46.640","Text":"Absolute value of a number squared is the same as the number squared."},{"Start":"01:46.640 ","End":"01:48.990","Text":"If it\u0027s positive, it\u0027s clearly true,"},{"Start":"01:48.990 ","End":"01:50.225","Text":"and if it\u0027s negative,"},{"Start":"01:50.225 ","End":"01:55.640","Text":"like absolute value of minus 4 squared is the same as 4 squared,"},{"Start":"01:55.640 ","End":"01:59.065","Text":"it\u0027s also the same as minus 4 in brackets squared."},{"Start":"01:59.065 ","End":"02:02.435","Text":"In any event it\u0027ll come out to be plus or 0."},{"Start":"02:02.435 ","End":"02:05.170","Text":"Now, absolute value of x less than k,"},{"Start":"02:05.170 ","End":"02:07.445","Text":"if we have this inequality,"},{"Start":"02:07.445 ","End":"02:13.665","Text":"then the solution is that x is between minus k and k,"},{"Start":"02:13.665 ","End":"02:15.240","Text":"it\u0027s well known,"},{"Start":"02:15.240 ","End":"02:20.630","Text":"and absolute value of x is bigger than k if and only if either x is"},{"Start":"02:20.630 ","End":"02:23.360","Text":"bigger than k or less than minus k."},{"Start":"02:23.360 ","End":"02:25.875","Text":"This is a reminder,"},{"Start":"02:25.875 ","End":"02:28.010","Text":"so we\u0027re not going to prove these."},{"Start":"02:28.010 ","End":"02:29.960","Text":"You should know these already."},{"Start":"02:29.960 ","End":"02:33.950","Text":"Finally, we can get to the triangle inequality."},{"Start":"02:33.950 ","End":"02:37.970","Text":"There\u0027s actually several versions."},{"Start":"02:37.970 ","End":"02:41.119","Text":"Any 1 of these, sometimes called a triangle inequality."},{"Start":"02:41.119 ","End":"02:44.930","Text":"But really the default form is the first 1."},{"Start":"02:44.930 ","End":"02:47.990","Text":"The absolute value of a plus b is less than or"},{"Start":"02:47.990 ","End":"02:51.755","Text":"equal to the absolute value of a plus absolute value of b."},{"Start":"02:51.755 ","End":"02:54.050","Text":"This is the triangle inequality,"},{"Start":"02:54.050 ","End":"02:56.420","Text":"even though the variants are also sometimes called that."},{"Start":"02:56.420 ","End":"02:59.350","Text":"You could replace the plus here with a minus,"},{"Start":"02:59.350 ","End":"03:05.360","Text":"and then there are 3 variants which are sometimes called reverse triangle inequality,"},{"Start":"03:05.360 ","End":"03:07.100","Text":"because instead of a less than or equal to,"},{"Start":"03:07.100 ","End":"03:09.610","Text":"we have a bigger than or equal to."},{"Start":"03:09.610 ","End":"03:13.460","Text":"Absolute value of a minus b is bigger or equal"},{"Start":"03:13.460 ","End":"03:16.925","Text":"to absolute value of a minus absolute value of b,"},{"Start":"03:16.925 ","End":"03:26.805","Text":"and we could switch a and b here because of this property."},{"Start":"03:26.805 ","End":"03:29.425","Text":"If we combine these 2,"},{"Start":"03:29.425 ","End":"03:32.420","Text":"we can say that the absolute value of a minus b is bigger or"},{"Start":"03:32.420 ","End":"03:36.610","Text":"equal to the absolute value of this."},{"Start":"03:36.610 ","End":"03:39.799","Text":"We can replace the minus here by a plus,"},{"Start":"03:39.799 ","End":"03:42.320","Text":"and it will also work."},{"Start":"03:42.320 ","End":"03:47.360","Text":"There\u0027s also a generalization of the first 1."},{"Start":"03:47.360 ","End":"03:48.800","Text":"Instead of 2 numbers,"},{"Start":"03:48.800 ","End":"03:50.780","Text":"if we have n numbers,"},{"Start":"03:50.780 ","End":"03:52.140","Text":"we\u0027ll call them a_1,"},{"Start":"03:52.140 ","End":"03:53.710","Text":"a_2 up to a_n,"},{"Start":"03:53.710 ","End":"03:59.875","Text":"the absolute value of the sum is less than or equal to the sum of the absolute values."},{"Start":"03:59.875 ","End":"04:04.055","Text":"We\u0027ll prove these in 1 of the exercises."},{"Start":"04:04.055 ","End":"04:09.335","Text":"You might be wondering about the reason for the name triangle inequality."},{"Start":"04:09.335 ","End":"04:12.185","Text":"It comes from the fact that in a triangle,"},{"Start":"04:12.185 ","End":"04:19.565","Text":"the sum of any 2 sides is bigger or equal to the lengths."},{"Start":"04:19.565 ","End":"04:23.405","Text":"The length of this plus the length of this is bigger or equal to the length of this."},{"Start":"04:23.405 ","End":"04:26.570","Text":"It\u0027s actually strictly bigger than if it\u0027s a proper triangle,"},{"Start":"04:26.570 ","End":"04:29.770","Text":"but there is a degenerate triangle which has been flattened,"},{"Start":"04:29.770 ","End":"04:32.350","Text":"and then it could be equals."},{"Start":"04:32.350 ","End":"04:39.635","Text":"In the plane, we take vectors and then it\u0027s not the absolute value, it\u0027s the norm."},{"Start":"04:39.635 ","End":"04:43.220","Text":"The norm is the length of a vector."},{"Start":"04:43.220 ","End":"04:46.590","Text":"That\u0027s the end of this clip."}],"ID":26702},{"Watched":false,"Name":"Bernoulli\u0027s Inequality","Duration":"1m 47s","ChapterTopicVideoID":25894,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Continuing with famous inequalities,"},{"Start":"00:03.570 ","End":"00:06.524","Text":"I count this as the third 1 in the series."},{"Start":"00:06.524 ","End":"00:09.045","Text":"We have Bernoulli\u0027s inequality."},{"Start":"00:09.045 ","End":"00:11.400","Text":"1 form of it is this,"},{"Start":"00:11.400 ","End":"00:19.470","Text":"that 1 plus x^n is bigger or equal to 1 plus nx for all natural numbers n,"},{"Start":"00:19.470 ","End":"00:24.405","Text":"provided that x is bigger or equal to minus 1."},{"Start":"00:24.405 ","End":"00:26.700","Text":"There\u0027s a variant of this;"},{"Start":"00:26.700 ","End":"00:34.065","Text":"1 minus x^n is less than or equal to 1 over 1 plus nx."},{"Start":"00:34.065 ","End":"00:36.300","Text":"This time x is more restricted,"},{"Start":"00:36.300 ","End":"00:39.790","Text":"it has to be in the close interval 0, 1."},{"Start":"00:39.790 ","End":"00:43.570","Text":"Now an example that n equals 3."},{"Start":"00:43.570 ","End":"00:48.635","Text":"We have that 1 plus x cubed is bigger or equal to 1 plus 3x,"},{"Start":"00:48.635 ","End":"00:50.960","Text":"provided x is bigger or equal to minus 1,"},{"Start":"00:50.960 ","End":"00:56.380","Text":"and 1 minus x cubed is less than or equal to 1 over 1 plus 3x,"},{"Start":"00:56.380 ","End":"00:58.540","Text":"if x is between 0 and 1."},{"Start":"00:58.540 ","End":"01:04.760","Text":"Let\u0027s even take a numerical example and that x equal 0.1,"},{"Start":"01:04.760 ","End":"01:13.095","Text":"then we get 1.1 cubed is bigger or equal to 1 plus 3 times 0.1 and that was 1.3."},{"Start":"01:13.095 ","End":"01:16.400","Text":"If you check it,1 plus 1 cubed is 1.331,"},{"Start":"01:16.400 ","End":"01:19.745","Text":"which really is bigger or equal to 1.3."},{"Start":"01:19.745 ","End":"01:27.945","Text":"Here,1 minus x is 0.9 cubed is less than or equal to here it\u0027s 1 over 1.3."},{"Start":"01:27.945 ","End":"01:33.305","Text":"On the calculator, it checks out that really this inequality holds."},{"Start":"01:33.305 ","End":"01:38.840","Text":"The proof for both of these can be found in the exercises belonging to"},{"Start":"01:38.840 ","End":"01:44.450","Text":"the section on proof by mathematical induction elsewhere in this chapter."},{"Start":"01:44.450 ","End":"01:48.390","Text":"It\u0027s there if you look for it, and we\u0027re done."}],"ID":26704},{"Watched":false,"Name":"Cauchy-Schwarz Inequality","Duration":"1m 44s","ChapterTopicVideoID":25895,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.080 ","End":"00:06.315","Text":"Next in our series of famous inequalities, the Cauchy-Schwarz Inequality."},{"Start":"00:06.315 ","End":"00:10.740","Text":"This actually has many versions in different branches of mathematics."},{"Start":"00:10.740 ","End":"00:12.990","Text":"For example, in linear algebra,"},{"Start":"00:12.990 ","End":"00:15.089","Text":"in something called inner product spaces."},{"Start":"00:15.089 ","End":"00:22.275","Text":"Here, we\u0027ll just express it to in a form involving 2 sets of n numbers, real numbers."},{"Start":"00:22.275 ","End":"00:24.419","Text":"We have 2n real numbers,"},{"Start":"00:24.419 ","End":"00:26.650","Text":"n here and n here."},{"Start":"00:26.650 ","End":"00:28.280","Text":"We get the following inequality."},{"Start":"00:28.280 ","End":"00:31.175","Text":"The first one says, multiply a_1 times b_1,"},{"Start":"00:31.175 ","End":"00:33.440","Text":"then a_2 times b_2,"},{"Start":"00:33.440 ","End":"00:36.380","Text":"and so on, and add them all up and square it."},{"Start":"00:36.380 ","End":"00:41.210","Text":"The other thing to do would be to take each one of these squared and add them,"},{"Start":"00:41.210 ","End":"00:43.610","Text":"and then each one of these squared and add them,"},{"Start":"00:43.610 ","End":"00:47.195","Text":"and then multiply the 2 sums of squares."},{"Start":"00:47.195 ","End":"00:52.160","Text":"If we do that, this will be less than or equal to this."},{"Start":"00:52.160 ","End":"00:55.795","Text":"It\u0027s not intuitive and let\u0027s give an example that might help."},{"Start":"00:55.795 ","End":"01:01.890","Text":"Will take the As to be 1, 3, 5,"},{"Start":"01:01.890 ","End":"01:08.860","Text":"and we\u0027ll take the Bs to be 2, 4, 6."},{"Start":"01:08.860 ","End":"01:11.915","Text":"Then we get 1 times 2,"},{"Start":"01:11.915 ","End":"01:13.655","Text":"plus 3 times 4,"},{"Start":"01:13.655 ","End":"01:15.410","Text":"plus 5 times 6,"},{"Start":"01:15.410 ","End":"01:19.220","Text":"all this squared is less than or equal to 1 squared plus 3 squared plus"},{"Start":"01:19.220 ","End":"01:23.825","Text":"5 squared times 2 squared plus 4 squared plus 6 squared."},{"Start":"01:23.825 ","End":"01:26.855","Text":"Let\u0027s actually see what it is numerically."},{"Start":"01:26.855 ","End":"01:29.420","Text":"This comes out to be 44 squared."},{"Start":"01:29.420 ","End":"01:32.060","Text":"This is 35 times 56,"},{"Start":"01:32.060 ","End":"01:33.740","Text":"and it checks out,"},{"Start":"01:33.740 ","End":"01:36.325","Text":"it really is less than or equal to."},{"Start":"01:36.325 ","End":"01:42.075","Text":"We\u0027ll prove this inequality in the exercises,"},{"Start":"01:42.075 ","End":"01:44.950","Text":"hence, that ends this clip."}],"ID":26705},{"Watched":false,"Name":"Exercise 1","Duration":"6m ","ChapterTopicVideoID":25896,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"This exercise is a preparation for the following exercise where we\u0027ll"},{"Start":"00:03.930 ","End":"00:08.190","Text":"prove that the geometric mean is less than or equal to the arithmetic mean."},{"Start":"00:08.190 ","End":"00:09.210","Text":"Get us 2 parts."},{"Start":"00:09.210 ","End":"00:14.220","Text":"In Part A, we have to show that if x and y satisfy x less than 1,"},{"Start":"00:14.220 ","End":"00:15.660","Text":"y bigger than 1,"},{"Start":"00:15.660 ","End":"00:18.865","Text":"then x plus y is bigger than x y plus 1."},{"Start":"00:18.865 ","End":"00:22.020","Text":"This in turn will be used to prove Part B,"},{"Start":"00:22.020 ","End":"00:23.250","Text":"which later as I say,"},{"Start":"00:23.250 ","End":"00:27.600","Text":"it will be used to prove the AM-GM inequality in Part B,"},{"Start":"00:27.600 ","End":"00:30.465","Text":"we\u0027ll show that if a_1 to a_2,"},{"Start":"00:30.465 ","End":"00:34.280","Text":"are positive numbers whose product is 1,"},{"Start":"00:34.280 ","End":"00:41.885","Text":"then the sum is bigger or equal to n. Let\u0027s start with Part A. X is less than 1,"},{"Start":"00:41.885 ","End":"00:43.850","Text":"so 1 minus x is positive,"},{"Start":"00:43.850 ","End":"00:45.290","Text":"y is bigger than 1,"},{"Start":"00:45.290 ","End":"00:47.479","Text":"so y minus 1 is positive,"},{"Start":"00:47.479 ","End":"00:50.415","Text":"positive times positive is positive."},{"Start":"00:50.415 ","End":"00:52.170","Text":"We get this. Now,"},{"Start":"00:52.170 ","End":"00:54.230","Text":"multiply it out we have this,"},{"Start":"00:54.230 ","End":"00:56.077","Text":"bring the minus 1 minus x, y"},{"Start":"00:56.077 ","End":"00:59.665","Text":"to the other side and we\u0027ve proven this."},{"Start":"00:59.665 ","End":"01:01.350","Text":"Let\u0027s get on to Part B,"},{"Start":"01:01.350 ","End":"01:07.470","Text":"which will do by induction on n. For each n,"},{"Start":"01:07.470 ","End":"01:09.780","Text":"we\u0027ll prove that for all a_1,"},{"Start":"01:09.780 ","End":"01:15.465","Text":"a_2 a_n which satisfy this, then this holds."},{"Start":"01:15.465 ","End":"01:17.655","Text":"Now, if n equals 1,"},{"Start":"01:17.655 ","End":"01:20.250","Text":"then we only have 1 member,"},{"Start":"01:20.250 ","End":"01:25.440","Text":"it\u0027s a_1 and it\u0027s equal to 1 because the product of a_1 is just a_1 is 1."},{"Start":"01:25.440 ","End":"01:29.010","Text":"So a_1 plus there\u0027s nothing to add,"},{"Start":"01:29.010 ","End":"01:31.200","Text":"so a_1 is bigger or equal to n, which is 1."},{"Start":"01:31.200 ","End":"01:32.670","Text":"That\u0027s clear. If a_1 is 1,"},{"Start":"01:32.670 ","End":"01:34.320","Text":"then a_1 is bigger or equal to 1."},{"Start":"01:34.320 ","End":"01:36.440","Text":"Let\u0027s do the induction part."},{"Start":"01:36.440 ","End":"01:38.360","Text":"First of all, the induction hypothesis."},{"Start":"01:38.360 ","End":"01:42.260","Text":"We\u0027ll fix some arbitrary n. For this n,"},{"Start":"01:42.260 ","End":"01:46.700","Text":"we know that if the product is 1,"},{"Start":"01:46.700 ","End":"01:51.470","Text":"then the sum is bigger or equal to n. From this will"},{"Start":"01:51.470 ","End":"01:57.020","Text":"show that it\u0027s true for n plus 1 that if we have a_1,"},{"Start":"01:57.020 ","End":"01:59.615","Text":"a_2 up to a_n plus 1,"},{"Start":"01:59.615 ","End":"02:00.860","Text":"whose product is 1,"},{"Start":"02:00.860 ","End":"02:04.205","Text":"then the sum will be bigger or equal to n plus 1."},{"Start":"02:04.205 ","End":"02:08.419","Text":"This arrow means that this part is the given,"},{"Start":"02:08.419 ","End":"02:12.150","Text":"and this part is what we\u0027re going to show."},{"Start":"02:12.290 ","End":"02:14.490","Text":"Divide into cases."},{"Start":"02:14.490 ","End":"02:15.650","Text":"In the first case,"},{"Start":"02:15.650 ","End":"02:17.570","Text":"which is the uninteresting case,"},{"Start":"02:17.570 ","End":"02:21.620","Text":"all the A\u0027s are equal to 1."},{"Start":"02:21.620 ","End":"02:23.705","Text":"Well, in this case,"},{"Start":"02:23.705 ","End":"02:25.835","Text":"if you add them all together,"},{"Start":"02:25.835 ","End":"02:27.590","Text":"1 plus 1 plus 1 plus 1,"},{"Start":"02:27.590 ","End":"02:30.080","Text":"but there\u0027s n plus 1 of them so we get n plus 1,"},{"Start":"02:30.080 ","End":"02:32.440","Text":"which is indeed bigger or equal to n plus 1."},{"Start":"02:32.440 ","End":"02:39.375","Text":"Now case 2, we\u0027ll assume that at least 1 of the a_i is bigger than 1."},{"Start":"02:39.375 ","End":"02:42.225","Text":"I claim that if 1 of the a_i is bigger than 1,"},{"Start":"02:42.225 ","End":"02:45.120","Text":"then another a_i well a_j is going to"},{"Start":"02:45.120 ","End":"02:48.080","Text":"be less than 1 because the product is going to be 1."},{"Start":"02:48.080 ","End":"02:51.290","Text":"If you make 1 bigger, you got to make 1 smaller. Well, let\u0027s see."},{"Start":"02:51.290 ","End":"02:53.135","Text":"Suppose on the contrary,"},{"Start":"02:53.135 ","End":"02:54.770","Text":"I mean proof by contradiction,"},{"Start":"02:54.770 ","End":"02:58.380","Text":"that for all the j from 1 to n plus 1,"},{"Start":"02:58.380 ","End":"03:01.200","Text":"then a_j is bigger or equal to 1."},{"Start":"03:01.200 ","End":"03:03.435","Text":"That\u0027s the negation of less than 1."},{"Start":"03:03.435 ","End":"03:07.090","Text":"In that case, if you look at a_1 up to a_n plus 1,"},{"Start":"03:07.090 ","End":"03:09.305","Text":"they\u0027re all bigger or equal to 1,"},{"Start":"03:09.305 ","End":"03:14.445","Text":"except that we know that a_i is strictly bigger than 1."},{"Start":"03:14.445 ","End":"03:16.370","Text":"If you multiply all these together,"},{"Start":"03:16.370 ","End":"03:19.580","Text":"we get strictly bigger than 1 and that\u0027s a contradiction"},{"Start":"03:19.580 ","End":"03:23.800","Text":"because the product has to equal exactly 1."},{"Start":"03:23.800 ","End":"03:28.750","Text":"We know that a_i is bigger than 1 and a_j is less than 1."},{"Start":"03:28.750 ","End":"03:35.825","Text":"Now, we can certainly rearrange the order of the elements a_1 to a_n plus 1."},{"Start":"03:35.825 ","End":"03:39.080","Text":"I mean, if we change the order of the product is still 1."},{"Start":"03:39.080 ","End":"03:40.820","Text":"If we change the order here,"},{"Start":"03:40.820 ","End":"03:42.890","Text":"the sum doesn\u0027t change production,"},{"Start":"03:42.890 ","End":"03:45.310","Text":"the sum don\u0027t change if you rearrange the order."},{"Start":"03:45.310 ","End":"03:49.230","Text":"We can push them to the end and assume that it\u0027s a_n"},{"Start":"03:49.230 ","End":"03:53.565","Text":"that\u0027s less than 1 and a_n plus 1 that\u0027s bigger than 1."},{"Start":"03:53.565 ","End":"03:57.690","Text":"Now, bunched the last 2 together like 1 single element and"},{"Start":"03:57.690 ","End":"04:01.730","Text":"then all together we have a product of n elements which is 1."},{"Start":"04:01.730 ","End":"04:04.820","Text":"Then we have that a_1 plus a_2 up to a_n,"},{"Start":"04:04.820 ","End":"04:06.875","Text":"a_n plus 1 as a single element,"},{"Start":"04:06.875 ","End":"04:09.920","Text":"this is bigger or equal to n. I\u0027m going to do some trick to"},{"Start":"04:09.920 ","End":"04:13.410","Text":"somehow split this product to a sum."},{"Start":"04:13.410 ","End":"04:15.885","Text":"That\u0027s where Part A will come in handy."},{"Start":"04:15.885 ","End":"04:20.385","Text":"At x equals a_n and y equals a_n plus 1,"},{"Start":"04:20.385 ","End":"04:23.880","Text":"then x is less than 1 and y is bigger than 1."},{"Start":"04:23.880 ","End":"04:30.320","Text":"We can use part a and get that x plus y is bigger than 1 plus x y."},{"Start":"04:30.320 ","End":"04:35.060","Text":"If we rearrange that x y is less than x plus y minus 1,"},{"Start":"04:35.060 ","End":"04:37.230","Text":"this translates to a_n,"},{"Start":"04:37.230 ","End":"04:41.290","Text":"a_n plus 1 less than a_n plus n plus 1 minus 1."},{"Start":"04:41.440 ","End":"04:44.615","Text":"If this is bigger or equal to n,"},{"Start":"04:44.615 ","End":"04:48.995","Text":"we replace this by something even bigger,"},{"Start":"04:48.995 ","End":"04:50.930","Text":"which is a_n plus n and plus or minus 1,"},{"Start":"04:50.930 ","End":"04:55.435","Text":"it\u0027ll be definitely bigger than n. We have the following."},{"Start":"04:55.435 ","End":"04:59.450","Text":"Now, just bring the 1 over to the other side and drop"},{"Start":"04:59.450 ","End":"05:03.710","Text":"the brackets and we have that the sum is bigger or equal to n plus 1."},{"Start":"05:03.710 ","End":"05:05.785","Text":"That\u0027s case 2."},{"Start":"05:05.785 ","End":"05:09.480","Text":"Case 3 is almost the same as case 2."},{"Start":"05:09.480 ","End":"05:12.390","Text":"Because if a_i is less than 1,"},{"Start":"05:12.390 ","End":"05:13.840","Text":"for some i,"},{"Start":"05:13.840 ","End":"05:15.605","Text":"I need to go back a moment."},{"Start":"05:15.605 ","End":"05:20.950","Text":"The opposite of all of them equaling 1 means that at least 1 of them is not equal to 1."},{"Start":"05:20.950 ","End":"05:24.250","Text":"We can break sub cases bigger than 1 and less than 1."},{"Start":"05:24.250 ","End":"05:26.900","Text":"This is the other possibility that\u0027s left,"},{"Start":"05:26.900 ","End":"05:32.510","Text":"a_1 less than 1 for some i and exactly the same as in case 2."},{"Start":"05:32.510 ","End":"05:34.520","Text":"We get that another a,"},{"Start":"05:34.520 ","End":"05:37.150","Text":"a_j is going to be bigger than 1."},{"Start":"05:37.150 ","End":"05:38.910","Text":"If 1 is bigger than 1 is less than,"},{"Start":"05:38.910 ","End":"05:41.040","Text":"if 1 is less then 1 is bigger."},{"Start":"05:41.040 ","End":"05:43.010","Text":"Then we can, just like before,"},{"Start":"05:43.010 ","End":"05:47.825","Text":"push them to the end and assume that it\u0027s a_n less than 1 and n plus 1 bigger than 1."},{"Start":"05:47.825 ","End":"05:49.640","Text":"Once we have this,"},{"Start":"05:49.640 ","End":"05:54.125","Text":"then we can just continue exactly the same as in case 2."},{"Start":"05:54.125 ","End":"05:55.850","Text":"There\u0027s nothing to do."},{"Start":"05:55.850 ","End":"05:57.515","Text":"We\u0027ve already done that."},{"Start":"05:57.515 ","End":"06:01.440","Text":"That concludes this exercise."}],"ID":26706},{"Watched":false,"Name":"Exercise 2","Duration":"2m 55s","ChapterTopicVideoID":25897,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.755","Text":"In this exercise, we\u0027re going to state and prove the inequality of the means."},{"Start":"00:04.755 ","End":"00:06.900","Text":"This is just to jog your memory."},{"Start":"00:06.900 ","End":"00:10.200","Text":"Harmonic mean, less than or equal to geometric mean,"},{"Start":"00:10.200 ","End":"00:12.195","Text":"less than or equal to arithmetic mean."},{"Start":"00:12.195 ","End":"00:14.385","Text":"Here\u0027s the statement,"},{"Start":"00:14.385 ","End":"00:18.585","Text":"x1 to xn are positive real numbers,"},{"Start":"00:18.585 ","End":"00:20.060","Text":"then the harmonic mean,"},{"Start":"00:20.060 ","End":"00:21.470","Text":"which is this expression,"},{"Start":"00:21.470 ","End":"00:23.645","Text":"is less than or equal to the geometric mean,"},{"Start":"00:23.645 ","End":"00:28.925","Text":"which is this, which is less than or equal to the arithmetic mean, which is this."},{"Start":"00:28.925 ","End":"00:35.315","Text":"Now the proof. We let a1 equal x1 over this."},{"Start":"00:35.315 ","End":"00:37.340","Text":"This is actually the geometric mean,"},{"Start":"00:37.340 ","End":"00:44.620","Text":"a2 is x2 over geometric mean up to an equals xn over geometric mean."},{"Start":"00:44.620 ","End":"00:48.695","Text":"We have n numbers and they\u0027re all positive."},{"Start":"00:48.695 ","End":"00:51.050","Text":"Let\u0027s multiply them out, a1,"},{"Start":"00:51.050 ","End":"00:53.750","Text":"a2 up to an is this."},{"Start":"00:53.750 ","End":"00:55.640","Text":"If we multiply it,"},{"Start":"00:55.640 ","End":"01:00.650","Text":"we get the square root of n to the power of n on the denominator and x1,"},{"Start":"01:00.650 ","End":"01:02.780","Text":"x2 up to xn on the numerator,"},{"Start":"01:02.780 ","End":"01:04.580","Text":"this comes out to be 1,"},{"Start":"01:04.580 ","End":"01:07.705","Text":"because the nth roots and the power of n cancel each other out."},{"Start":"01:07.705 ","End":"01:11.540","Text":"What we have is that the product of a1 to an is"},{"Start":"01:11.540 ","End":"01:15.560","Text":"1 and a1 to an are positive numbers so we can use"},{"Start":"01:15.560 ","End":"01:19.670","Text":"the previous exercise to conclude that the sum of these is bigger or"},{"Start":"01:19.670 ","End":"01:23.970","Text":"equal to n. That means that this plus this,"},{"Start":"01:23.970 ","End":"01:25.800","Text":"so 1 up to plus this,"},{"Start":"01:25.800 ","End":"01:29.750","Text":"bigger or equal to n. Give these a common denominator,"},{"Start":"01:29.750 ","End":"01:37.615","Text":"x1 plus x2 and so on over geometric mean is bigger or equal to n. Now rearrange a bit,"},{"Start":"01:37.615 ","End":"01:41.625","Text":"n over here, this over here and we have exactly that"},{"Start":"01:41.625 ","End":"01:45.875","Text":"the arithmetic mean is bigger or equal to the geometric mean."},{"Start":"01:45.875 ","End":"01:49.855","Text":"Yeah, or geometric mean is less than or equal to arithmetic, same thing."},{"Start":"01:49.855 ","End":"01:53.540","Text":"I will use this to prove the other part of the inequality that we\u0027re"},{"Start":"01:53.540 ","End":"01:57.080","Text":"missing which is that the harmonic is less than or equal to geometric."},{"Start":"01:57.080 ","End":"02:00.800","Text":"Here\u0027s the trick, replace each of these xi,"},{"Start":"02:00.800 ","End":"02:04.420","Text":"meaning x1 up to xn by its reciprocal."},{"Start":"02:04.420 ","End":"02:10.310","Text":"This is for any x1 to xn so these are still positive real numbers."},{"Start":"02:10.310 ","End":"02:13.370","Text":"What we get when we do the replacement is that"},{"Start":"02:13.370 ","End":"02:18.170","Text":"this is bigger or equal to the nth root of this product."},{"Start":"02:18.170 ","End":"02:23.170","Text":"Now we can multiply out and get 1 over the product of these,"},{"Start":"02:23.170 ","End":"02:26.055","Text":"I mean, actually take the 1 outside of the root."},{"Start":"02:26.055 ","End":"02:28.175","Text":"We\u0027re left with this."},{"Start":"02:28.175 ","End":"02:32.705","Text":"Next, we can take the reciprocal of an inequality if we invert it."},{"Start":"02:32.705 ","End":"02:36.115","Text":"For example, if 3 is bigger or equal to 2,"},{"Start":"02:36.115 ","End":"02:38.875","Text":"then 1/3 is less than or equal to a 1/2."},{"Start":"02:38.875 ","End":"02:45.065","Text":"Just take the reciprocal of this and the reciprocal of this,"},{"Start":"02:45.065 ","End":"02:47.030","Text":"which is just turn the 1 over it,"},{"Start":"02:47.030 ","End":"02:50.090","Text":"it comes to the numerator and this gives us that"},{"Start":"02:50.090 ","End":"02:53.660","Text":"the harmonic mean is less than or equal to geometric mean."},{"Start":"02:53.660 ","End":"02:56.760","Text":"That concludes this exercise."}],"ID":26707},{"Watched":false,"Name":"Exercise 3 part I","Duration":"5m 56s","ChapterTopicVideoID":25898,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.705","Text":"In this exercise, we\u0027ll prove several variations of the triangle inequality."},{"Start":"00:06.705 ","End":"00:09.040","Text":"Here, I\u0027ve got 6 of them."},{"Start":"00:09.040 ","End":"00:10.610","Text":"We\u0027ll start with part 1,"},{"Start":"00:10.610 ","End":"00:14.180","Text":"which is the basic triangle inequality."},{"Start":"00:14.180 ","End":"00:17.025","Text":"We\u0027ll prove it in 3 different ways."},{"Start":"00:17.025 ","End":"00:20.145","Text":"Let me just remind you what it is here."},{"Start":"00:20.145 ","End":"00:23.310","Text":"We want absolute value of a plus b less than or"},{"Start":"00:23.310 ","End":"00:26.490","Text":"equal to absolute value of a plus absolute value of b."},{"Start":"00:26.490 ","End":"00:31.270","Text":"Proof 1, note that any number like"},{"Start":"00:31.270 ","End":"00:35.790","Text":"a is between absolute value of a and minus absolute value of a."},{"Start":"00:35.790 ","End":"00:38.150","Text":"This is equal to 1 of these 2."},{"Start":"00:38.150 ","End":"00:40.850","Text":"If it\u0027s positive or 0, it\u0027s equal to this,"},{"Start":"00:40.850 ","End":"00:42.845","Text":"if it\u0027s negative it\u0027s equal to this,"},{"Start":"00:42.845 ","End":"00:45.585","Text":"fairly clear same thing with b."},{"Start":"00:45.585 ","End":"00:47.535","Text":"Then adding these 2,"},{"Start":"00:47.535 ","End":"00:50.135","Text":"this double inequality we have in the middle,"},{"Start":"00:50.135 ","End":"00:51.965","Text":"a plus b,"},{"Start":"00:51.965 ","End":"00:54.860","Text":"and then on the right we have absolute value of b plus"},{"Start":"00:54.860 ","End":"00:57.700","Text":"absolute value of a, and here minus."},{"Start":"00:57.700 ","End":"01:00.870","Text":"I\u0027m calling this k, I\u0027m calling this x,"},{"Start":"01:00.870 ","End":"01:07.460","Text":"because we\u0027re going to use the rule that absolute value of x less than k if and only if"},{"Start":"01:07.460 ","End":"01:10.760","Text":"x is between minus k and k. It works"},{"Start":"01:10.760 ","End":"01:14.615","Text":"also with less than or equal to just what we have here."},{"Start":"01:14.615 ","End":"01:19.005","Text":"We can conclude that this is true."},{"Start":"01:19.005 ","End":"01:20.990","Text":"Absolute value of x,"},{"Start":"01:20.990 ","End":"01:23.450","Text":"which is absolute value of a plus b,"},{"Start":"01:23.450 ","End":"01:28.935","Text":"is less than the absolute value of a plus absolute value of b. Yeah, that\u0027s the proof."},{"Start":"01:28.935 ","End":"01:31.205","Text":"Now proof number 2,"},{"Start":"01:31.205 ","End":"01:33.470","Text":"again, we\u0027re going to prove the same thing."},{"Start":"01:33.470 ","End":"01:37.880","Text":"Recall the rule that the absolute value of x squared is the same as x squared,"},{"Start":"01:37.880 ","End":"01:40.190","Text":"because whether x is positive or negative,"},{"Start":"01:40.190 ","End":"01:42.995","Text":"the sign drops, we just get a positive."},{"Start":"01:42.995 ","End":"01:48.550","Text":"We can say the absolute value of x plus y squared is the same as x plus y squared."},{"Start":"01:48.550 ","End":"01:51.180","Text":"Expand the brackets."},{"Start":"01:51.180 ","End":"01:52.725","Text":"Then these 2,"},{"Start":"01:52.725 ","End":"01:54.390","Text":"the first and last terms,"},{"Start":"01:54.390 ","End":"01:56.810","Text":"can be written as absolute value of x squared"},{"Start":"01:56.810 ","End":"01:59.935","Text":"absolute value of y squared again, using this rule."},{"Start":"01:59.935 ","End":"02:05.465","Text":"The middle bit, we can take the absolute value and we can only increase it."},{"Start":"02:05.465 ","End":"02:12.480","Text":"We have this middle term breaks up as twice absolute value of x absolute value of y."},{"Start":"02:12.650 ","End":"02:18.125","Text":"What we have is that this is less than or equal to this squared."},{"Start":"02:18.125 ","End":"02:20.570","Text":"Now, for positive number"},{"Start":"02:20.570 ","End":"02:23.195","Text":"squared is less than or equal to another positive number squared,"},{"Start":"02:23.195 ","End":"02:25.625","Text":"then the numbers themselves are the same,"},{"Start":"02:25.625 ","End":"02:27.485","Text":"less than or equal to."},{"Start":"02:27.485 ","End":"02:30.970","Text":"This is the proof number 2."},{"Start":"02:30.970 ","End":"02:34.130","Text":"Now proof number 3 of this,"},{"Start":"02:34.130 ","End":"02:37.065","Text":"which involves breaking up into cases."},{"Start":"02:37.065 ","End":"02:40.660","Text":"First, the case that 1 or the other is 0,"},{"Start":"02:40.660 ","End":"02:44.585","Text":"let\u0027s say a is 0 because it\u0027ll be the same if b is 0,"},{"Start":"02:44.585 ","End":"02:49.360","Text":"say a is 0, then absolute value of x plus absolute value of b."},{"Start":"02:49.360 ","End":"02:53.409","Text":"This is 0 and absolute value of 0 is 0."},{"Start":"02:53.409 ","End":"02:57.070","Text":"Now, this is just absolute value of b,"},{"Start":"02:57.070 ","End":"02:59.410","Text":"and b is equal to 0 plus b,"},{"Start":"02:59.410 ","End":"03:01.675","Text":"and 0 is equal to a."},{"Start":"03:01.675 ","End":"03:04.000","Text":"This is equal to this."},{"Start":"03:04.000 ","End":"03:07.430","Text":"If it\u0027s equal to this, It\u0027s less than or equal to."},{"Start":"03:07.470 ","End":"03:11.620","Text":"The next case will take us where they\u0027re both positive."},{"Start":"03:11.620 ","End":"03:13.870","Text":"If they\u0027re both positive,"},{"Start":"03:13.870 ","End":"03:18.875","Text":"then the absolute value of a is equal to a and same for b."},{"Start":"03:18.875 ","End":"03:23.945","Text":"This is positive, so it\u0027s equal to the absolute value of a plus b."},{"Start":"03:23.945 ","End":"03:25.865","Text":"This equals this."},{"Start":"03:25.865 ","End":"03:28.175","Text":"It\u0027s less than or equal to."},{"Start":"03:28.175 ","End":"03:31.650","Text":"This is less than or equal to this, well, it\u0027s equal to."},{"Start":"03:33.140 ","End":"03:39.795","Text":"The next case is where a is negative and b is positive,"},{"Start":"03:39.795 ","End":"03:43.010","Text":"and it will be entirely equivalent for the other way around because we"},{"Start":"03:43.010 ","End":"03:46.145","Text":"can always just switch a and b in the inequality."},{"Start":"03:46.145 ","End":"03:49.445","Text":"We\u0027ll prove that a negative and be positive."},{"Start":"03:49.445 ","End":"03:54.725","Text":"Now in general, the absolute value of x is either x or minus x,"},{"Start":"03:54.725 ","End":"03:56.285","Text":"whichever is the bigger of them."},{"Start":"03:56.285 ","End":"04:01.230","Text":"1 of them is positive and 1 of them\u0027s negative and the maximum is the positive 1."},{"Start":"04:01.340 ","End":"04:04.490","Text":"In particular for the x be a plus b,"},{"Start":"04:04.490 ","End":"04:10.330","Text":"absolute value of a plus b is the maximum of a plus b and minus a plus b."},{"Start":"04:10.330 ","End":"04:13.610","Text":"Now, because a is negative,"},{"Start":"04:13.610 ","End":"04:15.665","Text":"this is less than b,"},{"Start":"04:15.665 ","End":"04:21.530","Text":"and this which is minus a minus b is less than minus a,"},{"Start":"04:21.530 ","End":"04:23.775","Text":"because b is positive,"},{"Start":"04:23.775 ","End":"04:28.355","Text":"so minus b is negative and if you add b to it,"},{"Start":"04:28.355 ","End":"04:30.275","Text":"you only get bigger."},{"Start":"04:30.275 ","End":"04:33.800","Text":"Now the maximum of 2 positive numbers,"},{"Start":"04:33.800 ","End":"04:39.200","Text":"b and minus a is less than or equal to the sum."},{"Start":"04:39.200 ","End":"04:41.570","Text":"We can replace this,"},{"Start":"04:41.570 ","End":"04:43.370","Text":"the maximum by the sum,"},{"Start":"04:43.370 ","End":"04:46.390","Text":"so b plus minus a,"},{"Start":"04:46.390 ","End":"04:51.320","Text":"and this is equal to absolute value of b because b is positive,"},{"Start":"04:51.320 ","End":"04:55.250","Text":"and this is absolute value of a because a is negative."},{"Start":"04:55.250 ","End":"04:57.890","Text":"This is less than this."},{"Start":"04:57.890 ","End":"05:01.744","Text":"Yeah, we can switch the order and that\u0027s what we wanted."},{"Start":"05:01.744 ","End":"05:04.400","Text":"There\u0027s 1 more case we haven\u0027t considered,"},{"Start":"05:04.400 ","End":"05:07.055","Text":"a negative and b negative."},{"Start":"05:07.055 ","End":"05:09.440","Text":"In this case, the absolute value of"},{"Start":"05:09.440 ","End":"05:15.065","Text":"a plus the absolute value of b is equal to minus a plus minus b."},{"Start":"05:15.065 ","End":"05:17.360","Text":"Because when a number is negative,"},{"Start":"05:17.360 ","End":"05:20.810","Text":"the absolute value is minus that something."},{"Start":"05:20.810 ","End":"05:26.540","Text":"Now, this is equal to minus of a plus b,"},{"Start":"05:26.540 ","End":"05:30.920","Text":"and a plus b is negative because a is negative and b is negative,"},{"Start":"05:30.920 ","End":"05:32.765","Text":"negative plus negative is negative."},{"Start":"05:32.765 ","End":"05:36.440","Text":"Again, we can use this rule that if number is negative,"},{"Start":"05:36.440 ","End":"05:38.720","Text":"the absolute value is minus,"},{"Start":"05:38.720 ","End":"05:41.565","Text":"or the other way around minus it,"},{"Start":"05:41.565 ","End":"05:43.620","Text":"is the absolute value."},{"Start":"05:43.620 ","End":"05:45.885","Text":"This is equal to this."},{"Start":"05:45.885 ","End":"05:48.725","Text":"This is less than or equal to this."},{"Start":"05:48.725 ","End":"05:51.150","Text":"That\u0027s the last case."},{"Start":"05:51.150 ","End":"05:55.800","Text":"We\u0027re done with this part of the triangle inequality."}],"ID":26708},{"Watched":false,"Name":"Exercise 3 part II","Duration":"3m 29s","ChapterTopicVideoID":25889,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.805","Text":"Continuing with this exercise,"},{"Start":"00:02.805 ","End":"00:05.445","Text":"we just proved part 1,"},{"Start":"00:05.445 ","End":"00:07.290","Text":"and here we\u0027ll do 2,"},{"Start":"00:07.290 ","End":"00:09.690","Text":"3, 4, and 5 in this clip."},{"Start":"00:09.690 ","End":"00:11.730","Text":"Start with number 2."},{"Start":"00:11.730 ","End":"00:15.510","Text":"This is what we have to prove and we\u0027ll use number 1,"},{"Start":"00:15.510 ","End":"00:18.180","Text":"the basic inequality to do it. Here\u0027s how we do it."},{"Start":"00:18.180 ","End":"00:21.510","Text":"Write this as a plus minus b."},{"Start":"00:21.510 ","End":"00:25.050","Text":"Then by the regular triangle inequality,"},{"Start":"00:25.050 ","End":"00:29.865","Text":"this is less than or equal to absolute value of a plus absolute value of minus b,"},{"Start":"00:29.865 ","End":"00:33.300","Text":"this is what I\u0027m using, let\u0027s say, number 1 here."},{"Start":"00:33.300 ","End":"00:37.280","Text":"Absolute value of minus B is the same as absolute value of b."},{"Start":"00:37.280 ","End":"00:40.940","Text":"We\u0027ve shown that this is less than or equal to this,"},{"Start":"00:40.940 ","End":"00:42.925","Text":"which is what we had to show."},{"Start":"00:42.925 ","End":"00:45.935","Text":"Number 3, it\u0027s actually 2 parts."},{"Start":"00:45.935 ","End":"00:47.975","Text":"Start with this part."},{"Start":"00:47.975 ","End":"00:53.720","Text":"Now, absolute value of a can be written as absolute value of a minus b plus b."},{"Start":"00:53.720 ","End":"00:57.860","Text":"Then we can use the triangle inequality, the regular 1."},{"Start":"00:57.860 ","End":"01:00.740","Text":"Say this is less than or equal to absolute value of this,"},{"Start":"01:00.740 ","End":"01:02.480","Text":"absolute value of this."},{"Start":"01:02.480 ","End":"01:05.320","Text":"Yeah, I just wrote it here, that\u0027s what I\u0027m using."},{"Start":"01:05.320 ","End":"01:09.080","Text":"Now you can bring the absolute value of b over to"},{"Start":"01:09.080 ","End":"01:13.535","Text":"the other side and then switch the sides and we get this,"},{"Start":"01:13.535 ","End":"01:16.595","Text":"which is the first part of these 2,"},{"Start":"01:16.595 ","End":"01:18.395","Text":"skip to the second half."},{"Start":"01:18.395 ","End":"01:20.870","Text":"First of all, if it\u0027s true for any a and b,"},{"Start":"01:20.870 ","End":"01:23.315","Text":"I can write b instead of a and a instead of b."},{"Start":"01:23.315 ","End":"01:24.620","Text":"That\u0027s got to also be true,"},{"Start":"01:24.620 ","End":"01:26.870","Text":"is nothing special about the letters a and b."},{"Start":"01:26.870 ","End":"01:28.734","Text":"This is also true."},{"Start":"01:28.734 ","End":"01:32.480","Text":"Now we know that the absolute value of b minus a is the same as"},{"Start":"01:32.480 ","End":"01:36.920","Text":"the absolute value of a minus b or x minus y is y minus x."},{"Start":"01:36.920 ","End":"01:40.025","Text":"We can replace this by this and keep this the same."},{"Start":"01:40.025 ","End":"01:44.055","Text":"Now we have the right-hand part of this pair."},{"Start":"01:44.055 ","End":"01:47.045","Text":"Now the next one, which is this,"},{"Start":"01:47.045 ","End":"01:50.735","Text":"I claim we can derive 4 from 3 from these 2."},{"Start":"01:50.735 ","End":"01:52.055","Text":"How do we do that?"},{"Start":"01:52.055 ","End":"01:56.990","Text":"Well, we showed that absolute value of a minus b is bigger or equal to"},{"Start":"01:56.990 ","End":"01:59.705","Text":"absolute value of a minus absolute value of b"},{"Start":"01:59.705 ","End":"02:04.330","Text":"and we\u0027ve got the same thing if we reverse these 2."},{"Start":"02:04.330 ","End":"02:06.220","Text":"I\u0027ll call this k,"},{"Start":"02:06.220 ","End":"02:08.540","Text":"the absolute value of a minus b."},{"Start":"02:08.540 ","End":"02:16.190","Text":"We have that absolute value of a minus b is bigger or equal to this,"},{"Start":"02:16.190 ","End":"02:19.510","Text":"which I\u0027ll call x to switch sides here and we have this,"},{"Start":"02:19.510 ","End":"02:22.265","Text":"I\u0027ve colored it so you can follow."},{"Start":"02:22.265 ","End":"02:29.165","Text":"This is bigger or equal to this with multiply this inequality by minus 1."},{"Start":"02:29.165 ","End":"02:33.795","Text":"That switches the order of the absolute value of a and absolute value of b,"},{"Start":"02:33.795 ","End":"02:40.025","Text":"minus k less than or equal to x less than or equal to k. When we have that,"},{"Start":"02:40.025 ","End":"02:43.760","Text":"then that\u0027s equivalent to saying that"},{"Start":"02:43.760 ","End":"02:47.825","Text":"the absolute value of x is less than k. Absolute value of x is this,"},{"Start":"02:47.825 ","End":"02:50.150","Text":"and k is this."},{"Start":"02:50.150 ","End":"02:52.040","Text":"We get what we wanted."},{"Start":"02:52.040 ","End":"02:56.670","Text":"That\u0027s number 4. Now number 5,"},{"Start":"02:56.670 ","End":"02:58.140","Text":"which is like number 4,"},{"Start":"02:58.140 ","End":"03:00.580","Text":"but with a plus here."},{"Start":"03:00.580 ","End":"03:06.410","Text":"What we can do is say a plus b is a plus minus b."},{"Start":"03:06.410 ","End":"03:08.195","Text":"Then using part 4,"},{"Start":"03:08.195 ","End":"03:10.460","Text":"this is bigger or equal to the absolute value of"},{"Start":"03:10.460 ","End":"03:14.135","Text":"absolute value of a minus absolute value of minus b."},{"Start":"03:14.135 ","End":"03:18.280","Text":"However, absolute value of minus b is the same as the absolute value of b."},{"Start":"03:18.280 ","End":"03:20.090","Text":"It\u0027s one of those rules of absolute value."},{"Start":"03:20.090 ","End":"03:24.200","Text":"We get that this is bigger or equal to this,"},{"Start":"03:24.200 ","End":"03:25.970","Text":"and that\u0027s what we had to show."},{"Start":"03:25.970 ","End":"03:29.800","Text":"The last part, number 6 will be in the following clip."}],"ID":26699},{"Watched":false,"Name":"Exercise 3 part III","Duration":"1m 33s","ChapterTopicVideoID":25890,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.029","Text":"Continuing this multi-part exercise,"},{"Start":"00:03.029 ","End":"00:07.110","Text":"all we have left is the last part number 6,"},{"Start":"00:07.110 ","End":"00:13.575","Text":"which is a generalized triangle inequality for the sum of n terms."},{"Start":"00:13.575 ","End":"00:20.055","Text":"We\u0027ll do this one by induction on n. If n equals 1,"},{"Start":"00:20.055 ","End":"00:23.460","Text":"then we just have absolute value of a_1"},{"Start":"00:23.460 ","End":"00:27.480","Text":"less than or equal to absolute value of a_1, and that\u0027s clear."},{"Start":"00:27.480 ","End":"00:31.980","Text":"Let\u0027s assume we\u0027ve proved it for some n. The induction"},{"Start":"00:31.980 ","End":"00:37.730","Text":"hypothesis that this is less than or equal to this for all a_1 to a_n."},{"Start":"00:37.730 ","End":"00:39.935","Text":"But the n is fixed."},{"Start":"00:39.935 ","End":"00:43.145","Text":"We have to prove it for n plus 1,"},{"Start":"00:43.145 ","End":"00:46.745","Text":"that for any a_1 through a_n plus 1,"},{"Start":"00:46.745 ","End":"00:49.954","Text":"this absolute value is less than or equal to this."},{"Start":"00:49.954 ","End":"00:53.865","Text":"What we\u0027ll do is we\u0027ll break this up as a sum of 2 things,"},{"Start":"00:53.865 ","End":"00:58.255","Text":"and use the triangle inequality, this one."},{"Start":"00:58.255 ","End":"01:01.610","Text":"We have that the absolute value of this plus this"},{"Start":"01:01.610 ","End":"01:08.510","Text":"is less than or equal to the absolute value of this plus the absolute value of this."},{"Start":"01:08.510 ","End":"01:11.810","Text":"Now, here we have only n terms."},{"Start":"01:11.810 ","End":"01:15.295","Text":"We can use the induction hypothesis,"},{"Start":"01:15.295 ","End":"01:20.525","Text":"and get that this is less than or equal to the sum of the absolute values here plus this."},{"Start":"01:20.525 ","End":"01:26.045","Text":"Now I will just throw out the brackets and we get the sum from a_1 up to a_n plus 1,"},{"Start":"01:26.045 ","End":"01:28.615","Text":"each one an absolute value."},{"Start":"01:28.615 ","End":"01:31.935","Text":"That concludes Part 6,"},{"Start":"01:31.935 ","End":"01:34.360","Text":"it\u0027s he last part."}],"ID":26700},{"Watched":false,"Name":"Exercise 4","Duration":"3m 25s","ChapterTopicVideoID":25891,"CourseChapterTopicPlaylistID":246312,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this exercise, we have to state and prove"},{"Start":"00:02.610 ","End":"00:05.670","Text":"the Cauchy-Schwarz inequality. That\u0027s part a."},{"Start":"00:05.670 ","End":"00:12.375","Text":"Part b we\u0027ll use part a to prove that the sum of a_i from 1 to n is 1,"},{"Start":"00:12.375 ","End":"00:15.570","Text":"then the sum of a_i squared from 1 to n is"},{"Start":"00:15.570 ","End":"00:19.275","Text":"bigger or equal to 1 over n. Start with part a,"},{"Start":"00:19.275 ","End":"00:22.860","Text":"which is, first of all, state the inequality."},{"Start":"00:22.860 ","End":"00:30.560","Text":"If we have numbers a_1 to a_n and b_1 to b_n real numbers, then I won\u0027t read it out."},{"Start":"00:30.560 ","End":"00:32.630","Text":"The following inequality holds."},{"Start":"00:32.630 ","End":"00:34.325","Text":"Let\u0027s prove it."},{"Start":"00:34.325 ","End":"00:36.754","Text":"If x is any real number,"},{"Start":"00:36.754 ","End":"00:38.180","Text":"then the following expression,"},{"Start":"00:38.180 ","End":"00:41.420","Text":"something squared plus something squared and so on is bigger or equal to 0."},{"Start":"00:41.420 ","End":"00:43.520","Text":"These a_1, a_2, and b_1, b_2,"},{"Start":"00:43.520 ","End":"00:46.675","Text":"etc, are taken from this list."},{"Start":"00:46.675 ","End":"00:50.805","Text":"Expanding each 1 as a quadratic,"},{"Start":"00:50.805 ","End":"00:55.350","Text":"a_1 squared x squared plus 2a_1b_1x would be 1 squared and so on."},{"Start":"00:55.350 ","End":"00:57.710","Text":"Sum is bigger or equal to 0."},{"Start":"00:57.710 ","End":"01:01.820","Text":"Now let\u0027s collect the terms of the x squared and the x and the free term."},{"Start":"01:01.820 ","End":"01:08.135","Text":"We have this and we\u0027ll label the coefficient of x squared as A,"},{"Start":"01:08.135 ","End":"01:13.550","Text":"here B, and here C. Let\u0027s divide it into 2 cases."},{"Start":"01:13.550 ","End":"01:16.125","Text":"If only a_i\u0027s are 0,"},{"Start":"01:16.125 ","End":"01:22.420","Text":"then the inequality holds because each of these is 0 and this is going to be 0."},{"Start":"01:22.420 ","End":"01:25.190","Text":"So we\u0027re going to have 0 less than or equal to 0."},{"Start":"01:25.190 ","End":"01:26.860","Text":"That\u0027s not an interesting case,"},{"Start":"01:26.860 ","End":"01:28.000","Text":"but the inequality holds."},{"Start":"01:28.000 ","End":"01:29.890","Text":"Assume that they\u0027re not all 0."},{"Start":"01:29.890 ","End":"01:31.960","Text":"In that case, A is not 0."},{"Start":"01:31.960 ","End":"01:33.100","Text":"In fact, it\u0027s positive."},{"Start":"01:33.100 ","End":"01:37.690","Text":"So we have a parabola function on the left."},{"Start":"01:38.000 ","End":"01:42.580","Text":"The quadratic has 0 or 1 root."},{"Start":"01:42.580 ","End":"01:44.440","Text":"If it\u0027s always bigger or equal to 0,"},{"Start":"01:44.440 ","End":"01:47.110","Text":"it doesn\u0027t cross the axis."},{"Start":"01:47.110 ","End":"01:49.329","Text":"In fact, I\u0027ll show you a picture."},{"Start":"01:49.329 ","End":"01:55.680","Text":"It\u0027s got to be either totally above the axis or just grazing the axis."},{"Start":"01:55.680 ","End":"01:57.545","Text":"But it can never be negative."},{"Start":"01:57.545 ","End":"01:59.645","Text":"So it\u0027s 1 of these 2 cases,"},{"Start":"01:59.645 ","End":"02:06.785","Text":"which means that the discriminant B squared minus 4AC is less than or equal to 0."},{"Start":"02:06.785 ","End":"02:08.825","Text":"This is the case where it equals 0."},{"Start":"02:08.825 ","End":"02:11.270","Text":"This is the case where it\u0027s less than 0,"},{"Start":"02:11.270 ","End":"02:14.275","Text":"where the quadratic has no real roots."},{"Start":"02:14.275 ","End":"02:18.545","Text":"From this we get that B squared is less than or equal to 4AC."},{"Start":"02:18.545 ","End":"02:20.690","Text":"If we write that out with B,"},{"Start":"02:20.690 ","End":"02:23.155","Text":"A, and C, from here we get this."},{"Start":"02:23.155 ","End":"02:26.900","Text":"All we have to do is divide by 4 here and by 2 squared,"},{"Start":"02:26.900 ","End":"02:28.205","Text":"which is 4 here,"},{"Start":"02:28.205 ","End":"02:30.800","Text":"and we get this is less than or equal to this."},{"Start":"02:30.800 ","End":"02:32.615","Text":"This is what we have to show."},{"Start":"02:32.615 ","End":"02:35.725","Text":"So that concludes part a of the exercise."},{"Start":"02:35.725 ","End":"02:40.065","Text":"In part b we have the sum of n numbers,"},{"Start":"02:40.065 ","End":"02:41.690","Text":"a_1 to a_n is 1."},{"Start":"02:41.690 ","End":"02:44.750","Text":"We have to show that the sum of the squares is bigger or equal to 1 over"},{"Start":"02:44.750 ","End":"02:48.835","Text":"n. We\u0027ll use the Cauchy-Schwarz inequality."},{"Start":"02:48.835 ","End":"02:51.780","Text":"a_1, a_2, a_n are as given."},{"Start":"02:51.780 ","End":"02:56.130","Text":"Let b_1 to b_n be all of them equal to 1."},{"Start":"02:56.130 ","End":"03:00.595","Text":"Apply the Cauchy-Schwarz inequality to the a\u0027s and the b\u0027s."},{"Start":"03:00.595 ","End":"03:07.425","Text":"This is what it is. Substitute all the b\u0027s to equal 1 and we have the following."},{"Start":"03:07.425 ","End":"03:10.940","Text":"1 squared plus 1 squared plus 1 squared is equal to"},{"Start":"03:10.940 ","End":"03:14.780","Text":"n. So we get that 1 squared is less than or equal to the sum of"},{"Start":"03:14.780 ","End":"03:20.210","Text":"the a_i squared times n. This thing is bigger or equal"},{"Start":"03:20.210 ","End":"03:26.920","Text":"to 1 over n. That\u0027s what we had to show. We\u0027re done."}],"ID":26701}],"Thumbnail":null,"ID":246312},{"Name":"The Real Number System","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"59s","ChapterTopicVideoID":25851,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.400","Text":"In this exercise, we\u0027re given that E is a non-empty subset of R,"},{"Start":"00:05.400 ","End":"00:08.280","Text":"which is bounded above, and as such,"},{"Start":"00:08.280 ","End":"00:12.450","Text":"it has a least upper bound or supremum."},{"Start":"00:12.450 ","End":"00:16.380","Text":"The idea of the exercise here is to show that the supremum is unique."},{"Start":"00:16.380 ","End":"00:19.230","Text":"In other words, if you have what looks like 2 of them,"},{"Start":"00:19.230 ","End":"00:20.940","Text":"Alpha and Beta,"},{"Start":"00:20.940 ","End":"00:23.100","Text":"then they\u0027re actually equal."},{"Start":"00:23.100 ","End":"00:26.920","Text":"Suprema is the plural of supremum, of course."},{"Start":"00:27.020 ","End":"00:30.525","Text":"The way we\u0027ll prove it is as follows."},{"Start":"00:30.525 ","End":"00:33.900","Text":"Notice that Alpha is a least upper bound,"},{"Start":"00:33.900 ","End":"00:35.235","Text":"and so is Beta,"},{"Start":"00:35.235 ","End":"00:38.370","Text":"but we\u0027ll just look at Beta as an upper bound."},{"Start":"00:38.370 ","End":"00:44.590","Text":"Certainly, the least upper bound is less than or equal to any of the upper bound."},{"Start":"00:44.590 ","End":"00:47.965","Text":"Similarly, vice versa,"},{"Start":"00:47.965 ","End":"00:51.080","Text":"Beta is less than or equal to Alpha."},{"Start":"00:51.080 ","End":"00:54.365","Text":"Now, if you have this inequality and this inequality,"},{"Start":"00:54.365 ","End":"01:00.300","Text":"then certainly the Alpha and Beta are equal. We\u0027re done."}],"ID":26655},{"Watched":false,"Name":"Exercise 2","Duration":"2m 17s","ChapterTopicVideoID":25852,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.190","Text":"In this exercise, we have to prove an inequality named after Bernoulli,"},{"Start":"00:05.190 ","End":"00:10.665","Text":"which says that if x is bigger than minus 1,"},{"Start":"00:10.665 ","End":"00:14.970","Text":"then 1 plus x to the power of n is bigger or equal to"},{"Start":"00:14.970 ","End":"00:22.770","Text":"1 plus nx for all natural numbers N. We\u0027ll take the natural numbers to exclude 0,"},{"Start":"00:22.770 ","End":"00:25.479","Text":"so 1, 2, 3, etc."},{"Start":"00:25.550 ","End":"00:28.650","Text":"We\u0027ll do a proof by induction."},{"Start":"00:28.650 ","End":"00:35.610","Text":"We\u0027ll name this proposition P of n. We have to prove P of 1."},{"Start":"00:35.610 ","End":"00:37.380","Text":"If P of k is true,"},{"Start":"00:37.380 ","End":"00:39.705","Text":"then P of k plus 1 is true."},{"Start":"00:39.705 ","End":"00:42.725","Text":"P of 1 is fairly easy."},{"Start":"00:42.725 ","End":"00:45.360","Text":"We just substitute n equals 1,"},{"Start":"00:45.360 ","End":"00:47.070","Text":"we have to prove this."},{"Start":"00:47.070 ","End":"00:52.340","Text":"Now, just slightly simplify it and we get 1 plus x bigger or equal to 1 plus x."},{"Start":"00:52.340 ","End":"00:53.570","Text":"Well, it\u0027s equal,"},{"Start":"00:53.570 ","End":"00:56.255","Text":"so it\u0027s bigger or equal to, so that\u0027s trivial."},{"Start":"00:56.255 ","End":"01:00.710","Text":"Now, the induction step from k to k plus 1,"},{"Start":"01:00.710 ","End":"01:06.730","Text":"we\u0027ll start off with P of k and we have to end up with P of k plus 1."},{"Start":"01:06.730 ","End":"01:12.980","Text":"We have this and then multiply both sides by 1 plus x."},{"Start":"01:12.980 ","End":"01:16.460","Text":"We\u0027ve got 1 plus x to k plus 1 which is this times this,"},{"Start":"01:16.460 ","End":"01:22.040","Text":"and we can multiply both sides because 1 plus x is positive. Why is that?"},{"Start":"01:22.040 ","End":"01:24.590","Text":"Because x is bigger than minus 1,"},{"Start":"01:24.590 ","End":"01:26.765","Text":"and that\u0027s why we need this condition."},{"Start":"01:26.765 ","End":"01:33.920","Text":"Multiply both sides by a positive number and then we can open the brackets,"},{"Start":"01:33.920 ","End":"01:39.600","Text":"we get 1 times 1 plus kx plus x which is k plus 1 x,"},{"Start":"01:39.600 ","End":"01:41.220","Text":"and then kx squared."},{"Start":"01:41.220 ","End":"01:46.100","Text":"Now, kx squared is bigger or"},{"Start":"01:46.100 ","End":"01:50.870","Text":"equal to 0 because k is positive and x squared is non-negative,"},{"Start":"01:50.870 ","End":"01:52.490","Text":"so that\u0027s bigger or equal to 0."},{"Start":"01:52.490 ","End":"01:55.490","Text":"If we drop this term,"},{"Start":"01:55.490 ","End":"01:58.985","Text":"we get something that\u0027s less than or equal to."},{"Start":"01:58.985 ","End":"02:01.775","Text":"Well, this is bigger or equal to this,"},{"Start":"02:01.775 ","End":"02:06.890","Text":"and that means that this is bigger or equal to this,"},{"Start":"02:06.890 ","End":"02:11.570","Text":"which is exactly what you get if you substitute n equals k plus"},{"Start":"02:11.570 ","End":"02:17.310","Text":"1 in the original proposition. We\u0027re done."}],"ID":26656},{"Watched":false,"Name":"Exercise 3","Duration":"1m 36s","ChapterTopicVideoID":25843,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"In this exercise, we have 2 real numbers,"},{"Start":"00:03.225 ","End":"00:05.250","Text":"Alpha and Beta,"},{"Start":"00:05.250 ","End":"00:08.009","Text":"where Alpha is smaller than Beta."},{"Start":"00:08.009 ","End":"00:12.750","Text":"In part a, we have to show that there is some natural number n,"},{"Start":"00:12.750 ","End":"00:16.005","Text":"such that if we add 1 over n to Alpha,"},{"Start":"00:16.005 ","End":"00:18.045","Text":"it\u0027s still smaller than Beta."},{"Start":"00:18.045 ","End":"00:19.830","Text":"In b very similarly,"},{"Start":"00:19.830 ","End":"00:22.695","Text":"we have to show that there is some natural number n,"},{"Start":"00:22.695 ","End":"00:26.265","Text":"such that if we subtract 1 over n from Beta,"},{"Start":"00:26.265 ","End":"00:28.155","Text":"it\u0027s still bigger than Alpha."},{"Start":"00:28.155 ","End":"00:33.700","Text":"In fact, to be the same end for part a and for part b ."},{"Start":"00:33.980 ","End":"00:38.240","Text":"We\u0027ll use the Archimedean property because"},{"Start":"00:38.240 ","End":"00:42.410","Text":"Beta minus Alpha is positive, follows from here."},{"Start":"00:42.410 ","End":"00:44.270","Text":"By the Archimedean property,"},{"Start":"00:44.270 ","End":"00:48.530","Text":"the sum integer that if we multiply it by Beta minus Alpha,"},{"Start":"00:48.530 ","End":"00:53.450","Text":"we get it to be bigger than any real number we want, say 1."},{"Start":"00:53.450 ","End":"00:54.965","Text":"If this is true,"},{"Start":"00:54.965 ","End":"00:58.340","Text":"then n times this is bigger than 1,"},{"Start":"00:58.340 ","End":"01:01.670","Text":"which means that Beta minus Alpha is bigger than 1"},{"Start":"01:01.670 ","End":"01:05.465","Text":"over n and we can deduce 2 things from here."},{"Start":"01:05.465 ","End":"01:11.000","Text":"We can deduce that Alpha plus 1 over n is less than Beta and we can also"},{"Start":"01:11.000 ","End":"01:17.420","Text":"deduce that Beta minus 1 over n is bigger than Alpha just by playing around with it."},{"Start":"01:17.420 ","End":"01:23.690","Text":"Now from this, we get that Alpha\u0027s less than Alpha plus 1 over n,"},{"Start":"01:23.690 ","End":"01:27.425","Text":"certainly because 1 over n is positive, less than Beta,"},{"Start":"01:27.425 ","End":"01:29.150","Text":"and from the second part,"},{"Start":"01:29.150 ","End":"01:32.645","Text":"we get what we needed in part b."},{"Start":"01:32.645 ","End":"01:36.840","Text":"That\u0027s all there is to it. We\u0027re done."}],"ID":26647},{"Watched":false,"Name":"Exercise 4","Duration":"2m 26s","ChapterTopicVideoID":25844,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.490","Text":"In this exercise, A is a non-empty set of real numbers."},{"Start":"00:05.490 ","End":"00:11.235","Text":"We\u0027re given that Alpha is an upper bound of A,"},{"Start":"00:11.235 ","End":"00:14.325","Text":"doesn\u0027t say least upper bound, just upper bound."},{"Start":"00:14.325 ","End":"00:18.705","Text":"Now, suppose that for every natural number N,"},{"Start":"00:18.705 ","End":"00:21.735","Text":"there is some element of A,"},{"Start":"00:21.735 ","End":"00:23.220","Text":"call it a_n,"},{"Start":"00:23.220 ","End":"00:27.510","Text":"which is bigger than Alpha minus 1 over n."},{"Start":"00:27.510 ","End":"00:32.690","Text":"Then we have to show that Alpha is the supremum of A,"},{"Start":"00:32.690 ","End":"00:36.060","Text":"meaning it\u0027s the least upper bound."},{"Start":"00:37.490 ","End":"00:40.445","Text":"Like I said, it\u0027s an upper bound."},{"Start":"00:40.445 ","End":"00:42.845","Text":"We just have to show that it\u0027s the least."},{"Start":"00:42.845 ","End":"00:44.525","Text":"If it\u0027s not the least,"},{"Start":"00:44.525 ","End":"00:46.670","Text":"then there\u0027s going to be a smaller one."},{"Start":"00:46.670 ","End":"00:49.190","Text":"What we\u0027re doing is a proof by contradiction."},{"Start":"00:49.190 ","End":"00:52.340","Text":"When we say suppose on the contrary then means we\u0027re going to go for"},{"Start":"00:52.340 ","End":"00:58.730","Text":"contradiction that we have a smaller upper bound than Alpha, call it Beta."},{"Start":"00:58.730 ","End":"01:02.180","Text":"Then we can choose a natural number N such that"},{"Start":"01:02.180 ","End":"01:06.875","Text":"1 over n is less than the positive number Alpha minus Beta."},{"Start":"01:06.875 ","End":"01:11.860","Text":"We did this in a previous exercise using the Archimedean property."},{"Start":"01:11.860 ","End":"01:16.520","Text":"From this we get that Beta is less than Alpha minus"},{"Start":"01:16.520 ","End":"01:22.650","Text":"1 over n. This Alpha minus n makes us look here,"},{"Start":"01:22.650 ","End":"01:27.340","Text":"and we can get an a_n a member of A,"},{"Start":"01:27.340 ","End":"01:29.660","Text":"which is bigger than Alpha minus"},{"Start":"01:29.660 ","End":"01:37.550","Text":"1 over n. Just copied this in reverse order."},{"Start":"01:37.550 ","End":"01:39.620","Text":"We get a less than."},{"Start":"01:39.620 ","End":"01:48.170","Text":"That means that Beta is less than a_n because Beta\u0027s less than Alpha minus 1 over n,"},{"Start":"01:48.170 ","End":"01:49.460","Text":"which is less than a_n,"},{"Start":"01:49.460 ","End":"01:53.645","Text":"so by the transitivity of the less than."},{"Start":"01:53.645 ","End":"02:01.910","Text":"Now, Beta is not an upper bound for a because one of the members of A,"},{"Start":"02:01.910 ","End":"02:04.280","Text":"at least one is bigger than Beta."},{"Start":"02:04.280 ","End":"02:06.410","Text":"If it\u0027s not the upper bound,"},{"Start":"02:06.410 ","End":"02:09.220","Text":"then we\u0027ve reached a contradiction."},{"Start":"02:09.220 ","End":"02:12.170","Text":"Where does the contradiction come from?"},{"Start":"02:12.170 ","End":"02:17.015","Text":"It\u0027s from assuming that there is an upper bound Beta which is less than Alpha."},{"Start":"02:17.015 ","End":"02:18.785","Text":"No such Beta exist,"},{"Start":"02:18.785 ","End":"02:21.045","Text":"there is none smaller than Alpha."},{"Start":"02:21.045 ","End":"02:26.500","Text":"Alpha is the smallest or least, and we\u0027re done."}],"ID":26648},{"Watched":false,"Name":"Exercise 5","Duration":"58s","ChapterTopicVideoID":25845,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, we\u0027re given E,"},{"Start":"00:03.000 ","End":"00:08.160","Text":"which is a non-empty subset of real numbers and it\u0027s bounded above."},{"Start":"00:08.160 ","End":"00:12.960","Text":"We\u0027re given that Alpha is not only an upper bound for E,"},{"Start":"00:12.960 ","End":"00:17.130","Text":"but it also belongs to E. In this case,"},{"Start":"00:17.130 ","End":"00:19.050","Text":"it turns out that Alpha is"},{"Start":"00:19.050 ","End":"00:24.045","Text":"the least upper bound of E and we\u0027re going to prove this by contradiction."},{"Start":"00:24.045 ","End":"00:25.730","Text":"That\u0027s how we phrase it."},{"Start":"00:25.730 ","End":"00:29.105","Text":"Suppose on the contrary that Alpha\u0027s not the least."},{"Start":"00:29.105 ","End":"00:30.965","Text":"Not the least, what does that mean?"},{"Start":"00:30.965 ","End":"00:32.555","Text":"Something smaller."},{"Start":"00:32.555 ","End":"00:36.295","Text":"There\u0027s an upper bound Beta which is less than Alpha."},{"Start":"00:36.295 ","End":"00:38.540","Text":"Now Alpha is in E,"},{"Start":"00:38.540 ","End":"00:40.145","Text":"we\u0027re given that here,"},{"Start":"00:40.145 ","End":"00:42.770","Text":"and Beta is less than Alpha,"},{"Start":"00:42.770 ","End":"00:46.730","Text":"so it can\u0027t be an upper bound for E because it has to be bigger or equal"},{"Start":"00:46.730 ","End":"00:51.090","Text":"to all the members of E and that\u0027s a contradiction."},{"Start":"00:51.090 ","End":"00:54.180","Text":"The contradiction came from supposing an Alpha\u0027s not least,"},{"Start":"00:54.180 ","End":"00:58.240","Text":"so it is least and we\u0027re done."}],"ID":26649},{"Watched":false,"Name":"Exercise 6","Duration":"3m 38s","ChapterTopicVideoID":25846,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we have any real number x."},{"Start":"00:03.300 ","End":"00:06.030","Text":"We\u0027re going to show 2 things about x."},{"Start":"00:06.030 ","End":"00:14.940","Text":"1, that there\u0027s an integer m such that x is between m and m plus 1, including m,"},{"Start":"00:14.940 ","End":"00:16.935","Text":"but excluding m plus 1,"},{"Start":"00:16.935 ","End":"00:21.360","Text":"and the other thing is that there\u0027s another integer, l,"},{"Start":"00:21.360 ","End":"00:27.720","Text":"or maybe the same integer such that x is between l and l plus 1, but this time,"},{"Start":"00:27.720 ","End":"00:30.240","Text":"inclusive of the l plus 1,"},{"Start":"00:30.240 ","End":"00:33.210","Text":"but not inclusive of l,"},{"Start":"00:33.210 ","End":"00:37.250","Text":"and we\u0027ll use the Archimedean property."},{"Start":"00:37.250 ","End":"00:43.180","Text":"There exists a natural number n such that n times 1, or n,"},{"Start":"00:43.180 ","End":"00:45.750","Text":"is bigger than x and likewise,"},{"Start":"00:45.750 ","End":"00:46.860","Text":"for minus x,"},{"Start":"00:46.860 ","End":"00:52.130","Text":"there\u0027s a natural number k such that k times 1 is bigger than minus x."},{"Start":"00:52.130 ","End":"00:56.815","Text":"If we take this and this and combine them, we get this."},{"Start":"00:56.815 ","End":"01:01.430","Text":"Now, we\u0027ll define a set of integers i,"},{"Start":"01:01.430 ","End":"01:08.005","Text":"all the integers i is less than or equal to n and bigger or equal to minus k,"},{"Start":"01:08.005 ","End":"01:12.425","Text":"and the claim is that this set is finite and non-empty."},{"Start":"01:12.425 ","End":"01:13.960","Text":"Well, it\u0027s certainly finite."},{"Start":"01:13.960 ","End":"01:17.210","Text":"There\u0027s only a finite number of integers in this range,"},{"Start":"01:17.210 ","End":"01:22.210","Text":"and the reason it\u0027s non-empty is that minus k belongs to it"},{"Start":"01:22.210 ","End":"01:27.950","Text":"because minus k is bigger or equal to minus k for sure and also,"},{"Start":"01:27.950 ","End":"01:32.820","Text":"minus k is less than x,"},{"Start":"01:32.820 ","End":"01:36.240","Text":"so less than or equal to x, also."},{"Start":"01:36.240 ","End":"01:37.920","Text":"If we have a finite,"},{"Start":"01:37.920 ","End":"01:39.260","Text":"non-empty set of integers,"},{"Start":"01:39.260 ","End":"01:40.910","Text":"it has a largest member,"},{"Start":"01:40.910 ","End":"01:42.380","Text":"and we\u0027ll call that m,"},{"Start":"01:42.380 ","End":"01:44.465","Text":"and this is the m here,"},{"Start":"01:44.465 ","End":"01:49.955","Text":"so the claim is that m less than or equal to x less than m plus 1."},{"Start":"01:49.955 ","End":"01:52.190","Text":"Now, the first part,"},{"Start":"01:52.190 ","End":"01:53.420","Text":"m less than or equal to x,"},{"Start":"01:53.420 ","End":"02:01.850","Text":"is clear because all the members of this set are less than or equal to x."},{"Start":"02:01.850 ","End":"02:08.960","Text":"The question is, why is x less than m plus 1?"},{"Start":"02:08.960 ","End":"02:10.895","Text":"That needs a bit of explaining."},{"Start":"02:10.895 ","End":"02:14.630","Text":"Let\u0027s do a quick mental proof by contradiction."},{"Start":"02:14.630 ","End":"02:16.550","Text":"If this isn\u0027t so,"},{"Start":"02:16.550 ","End":"02:20.830","Text":"then m plus 1 is less than or equal to x,"},{"Start":"02:20.830 ","End":"02:23.880","Text":"and if it\u0027s less than or equal to x,"},{"Start":"02:23.880 ","End":"02:26.300","Text":"then it\u0027s certainly less than or equal to n,"},{"Start":"02:26.300 ","End":"02:28.410","Text":"so that\u0027s this and also,"},{"Start":"02:28.410 ","End":"02:33.800","Text":"m plus 1 is bigger or equal to k because m is bigger or equal to k,"},{"Start":"02:33.800 ","End":"02:35.855","Text":"m is in this set."},{"Start":"02:35.855 ","End":"02:39.925","Text":"That would mean that m plus 1 satisfies this and this,"},{"Start":"02:39.925 ","End":"02:47.535","Text":"so m plus 1 would be inside this set and that contradicts the word largest,"},{"Start":"02:47.535 ","End":"02:51.160","Text":"m plus 1 can\u0027t be in this set,"},{"Start":"02:52.520 ","End":"02:55.365","Text":"so that proves part a."},{"Start":"02:55.365 ","End":"02:56.775","Text":"Now, on to b."},{"Start":"02:56.775 ","End":"02:59.835","Text":"We\u0027ll use a trick to use part a."},{"Start":"02:59.835 ","End":"03:04.895","Text":"We just let y be minus x and use part a,"},{"Start":"03:04.895 ","End":"03:08.700","Text":"so the sum integer m, natural number,"},{"Start":"03:08.700 ","End":"03:12.755","Text":"such that m less than or equal to y less than m plus 1,"},{"Start":"03:12.755 ","End":"03:16.310","Text":"and now we reverse this by multiplying"},{"Start":"03:16.310 ","End":"03:20.090","Text":"everything by minus and switch the direction of the inequality."},{"Start":"03:20.090 ","End":"03:23.480","Text":"Now, from this, if we just write it from right to left,"},{"Start":"03:23.480 ","End":"03:29.010","Text":"and we also replace l equals minus m minus 1,"},{"Start":"03:29.010 ","End":"03:30.090","Text":"so here we have l,"},{"Start":"03:30.090 ","End":"03:32.730","Text":"and here we have l plus 1,"},{"Start":"03:32.730 ","End":"03:36.320","Text":"so we get this, which is the second inequality,"},{"Start":"03:36.320 ","End":"03:38.790","Text":"and we are done."}],"ID":26650},{"Watched":false,"Name":"Exercise 7","Duration":"2m 36s","ChapterTopicVideoID":25847,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"For this exercise, we have to define the distance between a point and a set."},{"Start":"00:05.580 ","End":"00:07.425","Text":"You might be familiar with it."},{"Start":"00:07.425 ","End":"00:10.065","Text":"If we have a non-empty set,"},{"Start":"00:10.065 ","End":"00:14.130","Text":"capital A and a real number x,"},{"Start":"00:14.130 ","End":"00:16.980","Text":"we define the distance of x from A,"},{"Start":"00:16.980 ","End":"00:19.650","Text":"and this is the notation as follows."},{"Start":"00:19.650 ","End":"00:28.365","Text":"The distance from x to A is the infimum of all the distances from x to a point in A,"},{"Start":"00:28.365 ","End":"00:30.675","Text":"that\u0027s absolute value of x minus a,"},{"Start":"00:30.675 ","End":"00:35.220","Text":"and the infimum of this set, that\u0027s the distance."},{"Start":"00:35.220 ","End":"00:42.975","Text":"Now what we have to show is that if we have the least upper bound of the set,"},{"Start":"00:42.975 ","End":"00:48.395","Text":"then the distance of that least upper bound from the set is 0."},{"Start":"00:48.395 ","End":"00:52.620","Text":"By the way, it works for greatest lower bound also."},{"Start":"00:52.850 ","End":"00:55.880","Text":"We\u0027ll do a proof by contradiction."},{"Start":"00:55.880 ","End":"00:58.370","Text":"It\u0027s a contradiction or whenever it starts,"},{"Start":"00:58.370 ","End":"01:00.380","Text":"suppose on the contrary."},{"Start":"01:00.380 ","End":"01:02.930","Text":"If the distance is not 0,"},{"Start":"01:02.930 ","End":"01:06.170","Text":"it has to be bigger than 0 because it\u0027s certainly not negative,"},{"Start":"01:06.170 ","End":"01:08.720","Text":"as you can see from the definition."},{"Start":"01:08.720 ","End":"01:12.515","Text":"If it\u0027s bigger than 0, we can choose epsilon,"},{"Start":"01:12.515 ","End":"01:15.230","Text":"another positive number that\u0027s still smaller."},{"Start":"01:15.230 ","End":"01:18.534","Text":"For example, you could take 1/2 of this distance."},{"Start":"01:18.534 ","End":"01:25.415","Text":"Since Epsilon is less than the infimum of the set,"},{"Start":"01:25.415 ","End":"01:28.910","Text":"it\u0027s less than each of the elements of the set."},{"Start":"01:28.910 ","End":"01:32.225","Text":"In other words, it\u0027s less than each absolute value of x minus"},{"Start":"01:32.225 ","End":"01:38.000","Text":"a and so that explains this inequality."},{"Start":"01:38.000 ","End":"01:43.580","Text":"We can drop the absolute value because Alpha is bigger or equal to a."},{"Start":"01:43.580 ","End":"01:47.064","Text":"The reason that Alpha is bigger or equal to a,"},{"Start":"01:47.064 ","End":"01:49.430","Text":"it because Alpha is an upper bound,"},{"Start":"01:49.430 ","End":"01:51.110","Text":"it\u0027s the least upper bound, but in particular,"},{"Start":"01:51.110 ","End":"01:52.475","Text":"it\u0027s an upper bound of a,"},{"Start":"01:52.475 ","End":"01:56.015","Text":"so it\u0027s bigger or equal to each member of a."},{"Start":"01:56.015 ","End":"02:02.465","Text":"We have the Alpha minus epsilon is bigger than a for all a belonging to a,"},{"Start":"02:02.465 ","End":"02:07.310","Text":"to switch the a and the epsilon around."},{"Start":"02:07.310 ","End":"02:12.710","Text":"Now look, Alpha minus epsilon is bigger than each a,"},{"Start":"02:12.710 ","End":"02:14.689","Text":"so it\u0027s an upper bound,"},{"Start":"02:14.689 ","End":"02:18.350","Text":"but it\u0027s an upper bound of a which is less than Alpha,"},{"Start":"02:18.350 ","End":"02:21.290","Text":"in other words, less than the least upper bound."},{"Start":"02:21.290 ","End":"02:25.040","Text":"How can that be? The least is the least."},{"Start":"02:25.040 ","End":"02:32.060","Text":"That\u0027s a contradiction and that contradicts our assumption that this distance is not 0."},{"Start":"02:32.060 ","End":"02:37.230","Text":"It means that the distance is 0, and we\u0027re done."}],"ID":26651},{"Watched":false,"Name":"Exercise 8","Duration":"9m ","ChapterTopicVideoID":25848,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"This exercise has 4 parts,"},{"Start":"00:02.340 ","End":"00:04.245","Text":"we\u0027ll read each part as we come to it."},{"Start":"00:04.245 ","End":"00:08.880","Text":"In part a, we\u0027re given a positive real number whose square is"},{"Start":"00:08.880 ","End":"00:12.510","Text":"less than 2 and informally what we have to show is that"},{"Start":"00:12.510 ","End":"00:14.160","Text":"we can increase x a little bit"},{"Start":"00:14.160 ","End":"00:16.380","Text":"and the square will still be less than 2."},{"Start":"00:16.380 ","End":"00:20.040","Text":"Specifically, we increase x by 1 over n for sum n"},{"Start":"00:20.040 ","End":"00:21.795","Text":"and if n is big enough,"},{"Start":"00:21.795 ","End":"00:23.865","Text":"then this thing will hold."},{"Start":"00:23.865 ","End":"00:25.290","Text":"Similarly the other way around,"},{"Start":"00:25.290 ","End":"00:27.510","Text":"if x squared is bigger than 2,"},{"Start":"00:27.510 ","End":"00:30.045","Text":"then we can take a little bit off x."},{"Start":"00:30.045 ","End":"00:34.305","Text":"Specifically, we can take 1 over n from x,"},{"Start":"00:34.305 ","End":"00:39.780","Text":"and the square will still be bigger than 2. x squared is less than 2,"},{"Start":"00:39.780 ","End":"00:42.260","Text":"so let Epsilon be 2 minus x squared,"},{"Start":"00:42.260 ","End":"00:44.345","Text":"and obviously Epsilon is positive."},{"Start":"00:44.345 ","End":"00:47.580","Text":"Now note this inequality,"},{"Start":"00:47.580 ","End":"00:49.790","Text":"i f I square x plus 1 over n,"},{"Start":"00:49.790 ","End":"00:52.880","Text":"we get x squared plus 2x over n plus 1 over n"},{"Start":"00:52.880 ","End":"00:56.795","Text":"squared but we can replace 1 over n squared with something bigger,"},{"Start":"00:56.795 ","End":"00:58.100","Text":"namely 1 over n"},{"Start":"00:58.100 ","End":"01:00.295","Text":"and we get this inequality."},{"Start":"01:00.295 ","End":"01:02.715","Text":"By the Archimedean property,"},{"Start":"01:02.715 ","End":"01:07.610","Text":"we can find some n such that 2x plus 1 over n is less than Epsilon."},{"Start":"01:07.610 ","End":"01:09.290","Text":"Well really we should switch the n"},{"Start":"01:09.290 ","End":"01:10.640","Text":"and the Epsilon around that\u0027s how"},{"Start":"01:10.640 ","End":"01:14.180","Text":"we can find n bigger than 2x plus 1 over Epsilon,"},{"Start":"01:14.180 ","End":"01:17.155","Text":"just switch the n with the Epsilon."},{"Start":"01:17.155 ","End":"01:24.360","Text":"That gives us that x plus 1 over n squared is less than x squared plus 2x plus 1 over n,"},{"Start":"01:24.360 ","End":"01:29.600","Text":"so what we get is that x plus 1 over n squared is less than what\u0027s written here and"},{"Start":"01:29.600 ","End":"01:34.900","Text":"just combine these 2 to make it 2x plus 1 over n. Now,"},{"Start":"01:34.900 ","End":"01:37.289","Text":"2x plus 1 over n is less than Epsilon"},{"Start":"01:37.289 ","End":"01:40.790","Text":"so here we get x squared plus Epsilon, which is 2."},{"Start":"01:40.790 ","End":"01:45.250","Text":"In other words, x plus 1 over n squared is less than 2."},{"Start":"01:45.250 ","End":"01:46.625","Text":"That\u0027s the first part."},{"Start":"01:46.625 ","End":"01:51.245","Text":"Now let\u0027s do the other part for x squared bigger than 2, very similar."},{"Start":"01:51.245 ","End":"01:54.320","Text":"This case, let Epsilon be x squared minus 2,"},{"Start":"01:54.320 ","End":"01:56.120","Text":"so Epsilon is positive,"},{"Start":"01:56.120 ","End":"01:58.550","Text":"and in this case we get the inequality,"},{"Start":"01:58.550 ","End":"02:04.415","Text":"x minus 1 over n squared is bigger than x squared minus 2x over n,"},{"Start":"02:04.415 ","End":"02:09.470","Text":"it will be equal if I wrote plus 1 over n squared but if I remove this term,"},{"Start":"02:09.470 ","End":"02:11.605","Text":"then we get a bigger than."},{"Start":"02:11.605 ","End":"02:14.360","Text":"Again, using the Archimedean property,"},{"Start":"02:14.360 ","End":"02:17.690","Text":"we can find n such that 2x over n less than"},{"Start":"02:17.690 ","End":"02:23.130","Text":"Epsilon or really start with 2x over Epsilon less than n and then switch"},{"Start":"02:23.130 ","End":"02:27.680","Text":"the n and the Epsilon and that means that x minus 1 over n"},{"Start":"02:27.680 ","End":"02:32.845","Text":"squared is bigger than x squared minus 2x over n from here,"},{"Start":"02:32.845 ","End":"02:40.305","Text":"and that\u0027s bigger than x squared minus Epsilon because 2x over n is less than Epsilon,"},{"Start":"02:40.305 ","End":"02:45.745","Text":"so minus 2x over n is bigger than minus Epsilon."},{"Start":"02:45.745 ","End":"02:48.810","Text":"This squared is bigger than 2."},{"Start":"02:48.810 ","End":"02:50.640","Text":"Now part b."},{"Start":"02:50.640 ","End":"02:56.900","Text":"We have to show that this set A of all the positive rationals whose square is"},{"Start":"02:56.900 ","End":"03:01.160","Text":"less than 2 is bounded above in Q but"},{"Start":"03:01.160 ","End":"03:06.480","Text":"doesn\u0027t have its least upper bound in Q. Q is the rationals of course."},{"Start":"03:06.530 ","End":"03:10.070","Text":"What we can do is as an upper bound,"},{"Start":"03:10.070 ","End":"03:13.025","Text":"we can take 1-and-/2 for example."},{"Start":"03:13.025 ","End":"03:14.960","Text":"Suppose x is in A,"},{"Start":"03:14.960 ","End":"03:20.405","Text":"then x squared is less than 2 because of this,"},{"Start":"03:20.405 ","End":"03:22.595","Text":"and 2 is less than 2-and-1/4,"},{"Start":"03:22.595 ","End":"03:25.605","Text":"and because x is positive,"},{"Start":"03:25.605 ","End":"03:34.960","Text":"x has to be less than the square root of 2-and-1/4, which is 1-and-1/2."},{"Start":"03:34.960 ","End":"03:38.000","Text":"The second part does not have its least upper bound in Q,"},{"Start":"03:38.000 ","End":"03:41.110","Text":"we\u0027ll do by contradiction and we start,"},{"Start":"03:41.110 ","End":"03:49.620","Text":"suppose on the contrary that we have some m in Q which is a least upper bound of A."},{"Start":"03:49.620 ","End":"03:51.510","Text":"The 3 cases,"},{"Start":"03:51.510 ","End":"03:54.885","Text":"if m squared is equal to 2,"},{"Start":"03:54.885 ","End":"03:58.345","Text":"then it\u0027s a contradiction because there\u0027s a famous proof"},{"Start":"03:58.345 ","End":"04:02.480","Text":"that there\u0027s no rational square root of 2,"},{"Start":"04:02.480 ","End":"04:05.475","Text":"the Greeks discovered this."},{"Start":"04:05.475 ","End":"04:09.700","Text":"Next case is m squared bigger than 2."},{"Start":"04:09.700 ","End":"04:14.990","Text":"Now by part a, we can find something less than m,"},{"Start":"04:14.990 ","End":"04:17.345","Text":"specifically m minus 1 over n,"},{"Start":"04:17.345 ","End":"04:19.030","Text":"whose square is bigger than 2,"},{"Start":"04:19.030 ","End":"04:22.445","Text":"and notice that m minus 1 over n is"},{"Start":"04:22.445 ","End":"04:26.550","Text":"rational because m is rational and 1 over n is rational,"},{"Start":"04:26.550 ","End":"04:30.290","Text":"and so we have another upper bound which is"},{"Start":"04:30.290 ","End":"04:34.100","Text":"less than the least upper bound and that\u0027s a contradiction."},{"Start":"04:34.100 ","End":"04:35.920","Text":"What about the 3rd case?"},{"Start":"04:35.920 ","End":"04:37.650","Text":"If m squared is less than 2,"},{"Start":"04:37.650 ","End":"04:39.395","Text":"again by part a,"},{"Start":"04:39.395 ","End":"04:40.700","Text":"the first half of it,"},{"Start":"04:40.700 ","End":"04:45.830","Text":"we can find an answer to m plus 1 over n squared is less than 2."},{"Start":"04:45.830 ","End":"04:49.880","Text":"Here also, m plus 1 over n is"},{"Start":"04:49.880 ","End":"04:53.070","Text":"rational and its square is less than 2"},{"Start":"04:53.070 ","End":"04:56.780","Text":"and then it\u0027s bounded above by m,"},{"Start":"04:56.780 ","End":"05:00.890","Text":"because m is the least upper bound so m plus"},{"Start":"05:00.890 ","End":"05:05.405","Text":"1 over n has to be less than m and that\u0027s obviously a contradiction."},{"Start":"05:05.405 ","End":"05:10.430","Text":"That means that in all 3 cases we get a contradiction and so that"},{"Start":"05:10.430 ","End":"05:17.450","Text":"negates the supposition that there is a rational least upper bound of A."},{"Start":"05:17.450 ","End":"05:20.840","Text":"Now part c. We have to conclude from"},{"Start":"05:20.840 ","End":"05:25.685","Text":"part b that Q doesn\u0027t possess the least upper bound property."},{"Start":"05:25.685 ","End":"05:28.105","Text":"Just quickly remind you what b is,"},{"Start":"05:28.105 ","End":"05:32.390","Text":"we just showed that this set A is"},{"Start":"05:32.390 ","End":"05:37.370","Text":"bounded above in Q but doesn\u0027t have its least upper bound in Q,"},{"Start":"05:37.370 ","End":"05:40.525","Text":"and A is a subset of Q of course."},{"Start":"05:40.525 ","End":"05:44.055","Text":"We\u0027ve just shown that Q has a subset,"},{"Start":"05:44.055 ","End":"05:46.660","Text":"namely the A from part b,"},{"Start":"05:46.660 ","End":"05:48.365","Text":"which is bounded above,"},{"Start":"05:48.365 ","End":"05:52.345","Text":"but doesn\u0027t have a least upper bound in Q"},{"Start":"05:52.345 ","End":"05:57.970","Text":"and so by the definition of least upper bound property,"},{"Start":"05:57.970 ","End":"06:00.380","Text":"Q doesn\u0027t have it because if it did,"},{"Start":"06:00.380 ","End":"06:04.940","Text":"then all subsets would have a least upper bound in Q."},{"Start":"06:04.940 ","End":"06:07.540","Text":"Now part d,"},{"Start":"06:07.540 ","End":"06:11.090","Text":"which wants us to show that if Alpha"},{"Start":"06:11.090 ","End":"06:14.630","Text":"is the least upper bound of the set A we already defined,"},{"Start":"06:14.630 ","End":"06:17.900","Text":"then Alpha squared is equal to 2."},{"Start":"06:17.900 ","End":"06:22.485","Text":"We\u0027re going to do it by contradiction by supposing that Alpha squared is not 2."},{"Start":"06:22.485 ","End":"06:25.460","Text":"There\u0027s only 2 cases: It\u0027s either bigger than or smaller than."},{"Start":"06:25.460 ","End":"06:28.850","Text":"Let\u0027s start off with case bigger than."},{"Start":"06:28.850 ","End":"06:31.640","Text":"By part a, go back and look at it."},{"Start":"06:31.640 ","End":"06:34.670","Text":"You\u0027ll see that we can find a natural number"},{"Start":"06:34.670 ","End":"06:38.585","Text":"n such that Alpha minus 1 over n squared is bigger than 2."},{"Start":"06:38.585 ","End":"06:42.060","Text":"It\u0027s phrased with x not with Alpha, the same thing."},{"Start":"06:42.310 ","End":"06:46.055","Text":"Alpha minus 1 over n is less than Alpha,"},{"Start":"06:46.055 ","End":"06:50.300","Text":"and therefore it can\u0027t be an upper bound of A,"},{"Start":"06:50.300 ","End":"06:52.340","Text":"because Alpha is the least"},{"Start":"06:52.340 ","End":"06:55.310","Text":"and so it can\u0027t be another upper bound smaller than it."},{"Start":"06:55.310 ","End":"07:01.090","Text":"There exists some member of A which is bigger than Alpha."},{"Start":"07:01.090 ","End":"07:03.260","Text":"That\u0027s what it means not to be an upper bound,"},{"Start":"07:03.260 ","End":"07:09.710","Text":"is just at least 1 that it\u0027s not bigger or equal to and so a is bigger than Alpha minus 1"},{"Start":"07:09.710 ","End":"07:13.200","Text":"over n. Because I mean this is smaller than Alpha and a"},{"Start":"07:13.200 ","End":"07:16.985","Text":"is bigger than Alpha so we get a bunch of inequalities."},{"Start":"07:16.985 ","End":"07:21.830","Text":"From here, we get that 2 is less than Alpha minus 1 over n squared,"},{"Start":"07:21.830 ","End":"07:24.200","Text":"and from here, if we square both sides,"},{"Start":"07:24.200 ","End":"07:28.270","Text":"we get that Alpha minus 1 over n squared is less than a squared."},{"Start":"07:28.270 ","End":"07:34.760","Text":"a squared is less than 2 because a belongs to A and one of the defining conditions of"},{"Start":"07:34.760 ","End":"07:41.355","Text":"the set A is a squared is less than 2 for all a in A."},{"Start":"07:41.355 ","End":"07:46.445","Text":"We get a contradiction because it says here ultimately that 2 is less than 2."},{"Start":"07:46.445 ","End":"07:49.405","Text":"That takes care of that case."},{"Start":"07:49.405 ","End":"07:51.600","Text":"Now let\u0027s do the other case,"},{"Start":"07:51.600 ","End":"07:54.080","Text":"Alpha squared less than 2."},{"Start":"07:54.080 ","End":"07:55.220","Text":"Again, by Part a,"},{"Start":"07:55.220 ","End":"08:01.830","Text":"we can find a natural number such that Alpha plus 1 over n squared is still less than 2."},{"Start":"08:02.000 ","End":"08:08.240","Text":"You can always choose a rational number between any 2 real numbers that are different."},{"Start":"08:08.240 ","End":"08:12.005","Text":"We\u0027ll choose r between Alpha and Alpha plus 1 over n,"},{"Start":"08:12.005 ","End":"08:17.525","Text":"and r squared since r is less than Alpha plus 1 over n,"},{"Start":"08:17.525 ","End":"08:20.420","Text":"this is less than Alpha plus 1 over n squared,"},{"Start":"08:20.420 ","End":"08:22.885","Text":"which is less than 2 from here."},{"Start":"08:22.885 ","End":"08:25.470","Text":"r squired is less than 2 and it\u0027s"},{"Start":"08:25.470 ","End":"08:30.240","Text":"rational and that means that it belongs to A by the definition of"},{"Start":"08:30.240 ","End":"08:35.090","Text":"A. r has got to be less than or equal to"},{"Start":"08:35.090 ","End":"08:40.720","Text":"the least upper bound of A, which is Alpha."},{"Start":"08:40.720 ","End":"08:42.080","Text":"On the other hand,"},{"Start":"08:42.080 ","End":"08:46.025","Text":"it says here that Alpha is less than r,"},{"Start":"08:46.025 ","End":"08:48.035","Text":"meaning that r is bigger than Alpha,"},{"Start":"08:48.035 ","End":"08:51.040","Text":"so this contradicts this."},{"Start":"08:51.040 ","End":"08:54.650","Text":"We\u0027ve got a contradiction in both cases and it leads us"},{"Start":"08:54.650 ","End":"09:00.920","Text":"back to Alpha squared is equal to 2. We\u0027re done."}],"ID":26652},{"Watched":false,"Name":"Exercise 9","Duration":"5m 6s","ChapterTopicVideoID":25849,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"In this exercise, there are 3 parts,"},{"Start":"00:03.000 ","End":"00:04.950","Text":"but we start with a definition"},{"Start":"00:04.950 ","End":"00:08.715","Text":"and if we have a subset of real numbers, call it a,"},{"Start":"00:08.715 ","End":"00:17.760","Text":"then Minus a in words means you negate all the elements of a and formally,"},{"Start":"00:17.760 ","End":"00:19.440","Text":"set of all Minus x,"},{"Start":"00:19.440 ","End":"00:21.600","Text":"where x is in a."},{"Start":"00:21.600 ","End":"00:23.070","Text":"Now, here\u0027s the exercise."},{"Start":"00:23.070 ","End":"00:24.840","Text":"Suppose that we have a non-empty"},{"Start":"00:24.840 ","End":"00:27.720","Text":"bounded above subset of the reals,"},{"Start":"00:27.720 ","End":"00:32.440","Text":"call it S. We have to show 3 things about Minus S,"},{"Start":"00:32.440 ","End":"00:35.630","Text":"a, that it\u0027s bounded below B,"},{"Start":"00:35.630 ","End":"00:42.770","Text":"that the infimum of Minus S is Minus the supremum of S and just to remind,"},{"Start":"00:42.770 ","End":"00:45.740","Text":"the supremum is least upper bound and infimum is greatest"},{"Start":"00:45.740 ","End":"00:49.445","Text":"lower bound and we have to conclude from B,"},{"Start":"00:49.445 ","End":"00:53.360","Text":"then that the least upper bound property of"},{"Start":"00:53.360 ","End":"00:58.520","Text":"the reals implies the greatest lower bound property of the reals and vice versa."},{"Start":"00:58.520 ","End":"01:00.850","Text":"That was you have 1, you have the other."},{"Start":"01:00.850 ","End":"01:05.750","Text":"Let capital M be an upper bound for S. I mean,"},{"Start":"01:05.750 ","End":"01:08.210","Text":"we\u0027re told it\u0027s bounded above."},{"Start":"01:08.210 ","End":"01:15.690","Text":"Now the claim is that Minus M is a lower bound for Minus S. We have to"},{"Start":"01:15.690 ","End":"01:19.210","Text":"show that Minus M is less than or equal to every element of"},{"Start":"01:19.210 ","End":"01:23.335","Text":"Minus S. Take any x in Minus S,"},{"Start":"01:23.335 ","End":"01:27.280","Text":"by definition, Minus x is in S and by"},{"Start":"01:27.280 ","End":"01:32.495","Text":"definition of upper bound Minus x is less than or equal to M,"},{"Start":"01:32.495 ","End":"01:39.185","Text":"which means that x is bigger or equal to Minus M and that\u0027s what we wanted."},{"Start":"01:39.185 ","End":"01:42.375","Text":"Now part b, we have to show this equality."},{"Start":"01:42.375 ","End":"01:48.490","Text":"Let\u0027s label, this is beta and this is alpha as"},{"Start":"01:48.490 ","End":"01:51.850","Text":"here and we have to show that"},{"Start":"01:51.850 ","End":"01:57.300","Text":"alpha equals Minus beta or rather beta equals Minus alpha, same thing."},{"Start":"01:57.300 ","End":"02:01.980","Text":"Let\u0027s take an x in Minus S and as before,"},{"Start":"02:01.980 ","End":"02:05.265","Text":"that means that Minus x belongs to S."},{"Start":"02:05.265 ","End":"02:10.525","Text":"Minus x has to be less than or equal to the supremum."},{"Start":"02:10.525 ","End":"02:16.075","Text":"Multiply both sides by Minus 1 and we get x bigger or equal to Minus alpha."},{"Start":"02:16.075 ","End":"02:19.810","Text":"What we\u0027ve shown is for any x in Minus S,"},{"Start":"02:19.810 ","End":"02:22.595","Text":"x is bigger or equal to Minus alpha."},{"Start":"02:22.595 ","End":"02:28.150","Text":"Minus alpha is a lower bound of Minus S. Now as a lower bound,"},{"Start":"02:28.150 ","End":"02:31.120","Text":"it has to be less than or equal to the greatest"},{"Start":"02:31.120 ","End":"02:34.935","Text":"lower bound and the greatest lower bound is beta."},{"Start":"02:34.935 ","End":"02:40.635","Text":"We have an inequality 1 way now we\u0027ll prove the reverse inequality."},{"Start":"02:40.635 ","End":"02:45.015","Text":"Let\u0027s take y in S this time."},{"Start":"02:45.015 ","End":"02:51.375","Text":"Then Minus y is in Minus S. That means that"},{"Start":"02:51.375 ","End":"02:58.715","Text":"Minus y has to be bigger or equal to the least upper bound of Minus S,"},{"Start":"02:58.715 ","End":"03:00.035","Text":"and that\u0027s beta,"},{"Start":"03:00.035 ","End":"03:01.835","Text":"the infimum of Minus S,"},{"Start":"03:01.835 ","End":"03:08.464","Text":"multiply by Minus 1 and we\u0027ve got that y is less than or equal to Minus beta."},{"Start":"03:08.464 ","End":"03:14.450","Text":"Now this is true for all y in S. Minus beta is bigger or equal to the mole,"},{"Start":"03:14.450 ","End":"03:22.670","Text":"so Minus beta must be an upper bound for S and if it\u0027s an upper bound for S,"},{"Start":"03:22.670 ","End":"03:30.450","Text":"it has to be bigger or equal to the least upper bound and the least upper bound is alpha."},{"Start":"03:30.450 ","End":"03:35.415","Text":"That gives us the reverse inequality and I compare this and this,"},{"Start":"03:35.415 ","End":"03:37.290","Text":"the 1 hand less than or equal to,"},{"Start":"03:37.290 ","End":"03:42.330","Text":"on the 1 hand bigger or equal to, we get equality."},{"Start":"03:42.330 ","End":"03:44.090","Text":"Just to rephrase that,"},{"Start":"03:44.090 ","End":"03:49.510","Text":"the infimum of Minus S is Minus the supremum of S as required"},{"Start":"03:49.510 ","End":"03:56.465","Text":"and now part c. We assume that the reals have the least upper bound property."},{"Start":"03:56.465 ","End":"04:02.180","Text":"We have to show that the reals have the greatest lower bound property."},{"Start":"04:02.180 ","End":"04:05.630","Text":"What we have to show is that each non-empty bounded below"},{"Start":"04:05.630 ","End":"04:09.350","Text":"set has a greatest lower bound set,"},{"Start":"04:09.350 ","End":"04:12.030","Text":"meaning subset of the reals."},{"Start":"04:12.200 ","End":"04:17.625","Text":"Let\u0027s assume that S is non-empty and is bounded below."},{"Start":"04:17.625 ","End":"04:22.080","Text":"Minus S is also non-empty and I claim it\u0027s bounded"},{"Start":"04:22.080 ","End":"04:27.125","Text":"above because if big M is a lower bound for S,"},{"Start":"04:27.125 ","End":"04:31.610","Text":"then Minus M is an upper bound of Minus S. It\u0027s just like we did before."},{"Start":"04:31.610 ","End":"04:32.810","Text":"We just take an inequality,"},{"Start":"04:32.810 ","End":"04:35.615","Text":"multiply by Minus 1, it changes sign."},{"Start":"04:35.615 ","End":"04:37.980","Text":"That\u0027s easy to show."},{"Start":"04:38.440 ","End":"04:45.170","Text":"Minus S has a least upper bound because we\u0027re assuming the least upper bound property,"},{"Start":"04:45.170 ","End":"04:47.390","Text":"so bounded above and having an upper bound"},{"Start":"04:47.390 ","End":"04:50.375","Text":"means it has a least upper bound, call it alpha."},{"Start":"04:50.375 ","End":"04:51.725","Text":"By part b,"},{"Start":"04:51.725 ","End":"04:56.390","Text":"Minus alpha would be the greatest lower bound of Minus S,"},{"Start":"04:56.390 ","End":"05:03.115","Text":"which is of course just S and the other way around the vice versa is very similar,"},{"Start":"05:03.115 ","End":"05:05.105","Text":"no point in doing it again."},{"Start":"05:05.105 ","End":"05:07.560","Text":"Okay, we\u0027re done."}],"ID":26653},{"Watched":false,"Name":"Exercise 10","Duration":"5m 43s","ChapterTopicVideoID":25850,"CourseChapterTopicPlaylistID":246311,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"The main idea in this exercise is to show that"},{"Start":"00:03.870 ","End":"00:09.825","Text":"the square root of a whole number is either a whole number or it\u0027s irrational,"},{"Start":"00:09.825 ","End":"00:15.010","Text":"it can\u0027t just be a fraction without being a whole number."},{"Start":"00:15.140 ","End":"00:19.635","Text":"Let\u0027s start. K is a positive integer,"},{"Start":"00:19.635 ","End":"00:23.115","Text":"x is the square root of k; a real number."},{"Start":"00:23.115 ","End":"00:26.415","Text":"Suppose that x is rational,"},{"Start":"00:26.415 ","End":"00:32.700","Text":"that means we can write x in the form of m over n where m is an integer,"},{"Start":"00:32.700 ","End":"00:37.260","Text":"and n we can make it positive."},{"Start":"00:37.260 ","End":"00:40.520","Text":"If it\u0027s not positive just throw the minus up to the numerator,"},{"Start":"00:40.520 ","End":"00:43.715","Text":"and you can also take the least n possible."},{"Start":"00:43.715 ","End":"00:52.365","Text":"What that means is that n is the least positive integer such that nx is an integer,"},{"Start":"00:52.365 ","End":"00:59.040","Text":"in this case nx is m. It just means reduced and the denominator positive."},{"Start":"01:00.020 ","End":"01:10.415","Text":"Another integer n prime will equal n times x minus the floor function of x."},{"Start":"01:10.415 ","End":"01:13.400","Text":"If you don\u0027t remember what the floor function is,"},{"Start":"01:13.400 ","End":"01:16.595","Text":"basically this tells it all."},{"Start":"01:16.595 ","End":"01:21.290","Text":"It\u0027s an integer such that this inequality is satisfied."},{"Start":"01:21.290 ","End":"01:26.135","Text":"X is between this integer and the following integer."},{"Start":"01:26.135 ","End":"01:28.160","Text":"It could be the lower one,"},{"Start":"01:28.160 ","End":"01:30.240","Text":"but not the upper one."},{"Start":"01:31.340 ","End":"01:40.055","Text":"Part a; show that n prime is between 0 and n as written,"},{"Start":"01:40.055 ","End":"01:43.810","Text":"and that n prime times x is an integer."},{"Start":"01:43.810 ","End":"01:46.410","Text":"We\u0027ll start with a,"},{"Start":"01:46.410 ","End":"01:50.470","Text":"then we\u0027ll read b and c, and do them."},{"Start":"01:50.930 ","End":"01:54.215","Text":"From this bit here,"},{"Start":"01:54.215 ","End":"01:57.485","Text":"we can get that x minus"},{"Start":"01:57.485 ","End":"02:02.980","Text":"the floor function of x is bigger or equal to 0 and n is a positive integer,"},{"Start":"02:02.980 ","End":"02:05.755","Text":"so all of this is bigger or equal to 0."},{"Start":"02:05.755 ","End":"02:16.170","Text":"On the other hand from the second half of this inequality we get that this minus this;"},{"Start":"02:16.170 ","End":"02:24.130","Text":"I mean x minus the floor function of x is going to be less than 1, strictly less than."},{"Start":"02:24.500 ","End":"02:29.180","Text":"N prime times this is going to be less than"},{"Start":"02:29.180 ","End":"02:33.980","Text":"n. If we look at this and this and combine them,"},{"Start":"02:33.980 ","End":"02:41.435","Text":"so we get this which is what we had to show."},{"Start":"02:41.435 ","End":"02:44.035","Text":"We still have to show this part."},{"Start":"02:44.035 ","End":"02:50.645","Text":"Recall that x equals m over n. Now let\u0027s develop n prime x."},{"Start":"02:50.645 ","End":"02:58.135","Text":"This is equal to n times x minus floor x by definition."},{"Start":"02:58.135 ","End":"03:01.785","Text":"Expanding, we get nx squared"},{"Start":"03:01.785 ","End":"03:09.740","Text":"minus nx floor function of x. X squared is equal to k from here,"},{"Start":"03:09.740 ","End":"03:17.025","Text":"and so we get nk minus now nx is m. From here,"},{"Start":"03:17.025 ","End":"03:21.340","Text":"n times x is m. Altogether,"},{"Start":"03:21.340 ","End":"03:24.620","Text":"the main thing here is that this is an integer,"},{"Start":"03:24.620 ","End":"03:27.420","Text":"because everything here is an integer."},{"Start":"03:28.400 ","End":"03:32.055","Text":"That\u0027s the other part."},{"Start":"03:32.055 ","End":"03:34.410","Text":"We\u0027re done with Part a,"},{"Start":"03:34.410 ","End":"03:40.280","Text":"so on to Part b and we\u0027ll do this one by contradiction."},{"Start":"03:40.280 ","End":"03:44.735","Text":"We know that n prime is bigger or equal to 0,"},{"Start":"03:44.735 ","End":"03:49.280","Text":"so the contradiction would come if we suppose"},{"Start":"03:49.280 ","End":"03:54.875","Text":"that n prime not equal to 0 means bigger than 0."},{"Start":"03:54.875 ","End":"04:01.295","Text":"Now, n prime is less than n. We also showed that, here it is."},{"Start":"04:01.295 ","End":"04:06.285","Text":"N prime x is an integer also from Part a,"},{"Start":"04:06.285 ","End":"04:13.130","Text":"so n is not the least positive integer such that nx is an integer."},{"Start":"04:13.130 ","End":"04:15.000","Text":"We\u0027ve just found a smaller one;"},{"Start":"04:15.000 ","End":"04:19.950","Text":"n prime, which satisfies what it says here."},{"Start":"04:19.950 ","End":"04:22.015","Text":"Then let\u0027s highlight it."},{"Start":"04:22.015 ","End":"04:26.465","Text":"N is the least positive integer such that nx is an integer,"},{"Start":"04:26.465 ","End":"04:30.550","Text":"and we just found a smaller one; n prime."},{"Start":"04:30.550 ","End":"04:33.120","Text":"That\u0027s the contradiction,"},{"Start":"04:33.120 ","End":"04:37.845","Text":"and the contradiction means that n prime is 0."},{"Start":"04:37.845 ","End":"04:41.120","Text":"On to Part c,"},{"Start":"04:41.120 ","End":"04:47.180","Text":"x is root k. Now if x is rational,"},{"Start":"04:47.180 ","End":"04:51.730","Text":"we can write x equals m over n just as we did above."},{"Start":"04:51.730 ","End":"04:56.505","Text":"Then from Part b n prime;"},{"Start":"04:56.505 ","End":"04:59.205","Text":"yeah it\u0027s written here, is this."},{"Start":"04:59.205 ","End":"05:02.985","Text":"This is 0 but n is not 0,"},{"Start":"05:02.985 ","End":"05:09.750","Text":"so x is the floor function of x and that just means that x is an integer."},{"Start":"05:09.750 ","End":"05:16.069","Text":"We\u0027ve actually completed what we had to show because this statement here is logically"},{"Start":"05:16.069 ","End":"05:22.640","Text":"equivalent to saying that if root k is rational and not irrational,"},{"Start":"05:22.640 ","End":"05:26.915","Text":"then it must be the other case of the either or so it has to be"},{"Start":"05:26.915 ","End":"05:31.455","Text":"positive integer and we\u0027ve shown that."},{"Start":"05:31.455 ","End":"05:35.630","Text":"Well, the positive is clear because k"},{"Start":"05:35.630 ","End":"05:39.860","Text":"is positive so x has to be positive and it\u0027s an integer,"},{"Start":"05:39.860 ","End":"05:43.200","Text":"so it\u0027s a positive integer and we\u0027re done."}],"ID":26654}],"Thumbnail":null,"ID":246311}]

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1.1

1

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