Logic and Set Theory
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Operations on Sets
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Irrational Numbers
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[{"Name":"Logic and Set Theory","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Logical Connectives and Quantifiers","Duration":"4m 58s","ChapterTopicVideoID":24840,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24840.jpeg","UploadDate":"2021-06-23T01:52:50.3070000","DurationForVideoObject":"PT4M58S","Description":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":25608},{"Watched":false,"Name":"Exercise 1","Duration":"4m 4s","ChapterTopicVideoID":24844,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24844.jpeg","UploadDate":"2021-06-23T01:56:44.5700000","DurationForVideoObject":"PT4M4S","Description":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":25612},{"Watched":false,"Name":"Exercise 2","Duration":"1m 41s","ChapterTopicVideoID":24845,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24845.jpeg","UploadDate":"2021-06-23T01:57:06.6100000","DurationForVideoObject":"PT1M41S","Description":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":25613},{"Watched":false,"Name":"Sets and Elements","Duration":"4m 18s","ChapterTopicVideoID":24841,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24841.jpeg","UploadDate":"2021-06-23T01:53:42.4500000","DurationForVideoObject":"PT4M18S","Description":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":25609},{"Watched":false,"Name":"Exercise 3","Duration":"4m 32s","ChapterTopicVideoID":24831,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24831.jpeg","UploadDate":"2021-06-23T01:47:45.3300000","DurationForVideoObject":"PT4M32S","Description":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":25599},{"Watched":false,"Name":"Equality of Sets","Duration":"2m 40s","ChapterTopicVideoID":24830,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24830.jpeg","UploadDate":"2021-06-23T01:45:10.3870000","DurationForVideoObject":"PT2M40S","Description":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":25598},{"Watched":false,"Name":"Exercise 4","Duration":"1m 31s","ChapterTopicVideoID":24832,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24832.jpeg","UploadDate":"2021-06-23T01:48:03.3200000","DurationForVideoObject":"PT1M31S","Description":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":25600},{"Watched":false,"Name":"Finits, Infinite and Empty Sets","Duration":"3m 19s","ChapterTopicVideoID":24838,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24838.jpeg","UploadDate":"2021-06-23T01:51:05.3630000","DurationForVideoObject":"PT3M19S","Description":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":25606},{"Watched":false,"Name":"Subsets","Duration":"2m 50s","ChapterTopicVideoID":24843,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24843.jpeg","UploadDate":"2021-06-23T01:55:52.4270000","DurationForVideoObject":"PT2M50S","Description":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":25611},{"Watched":false,"Name":"Special Sets of Numbers","Duration":"7m 31s","ChapterTopicVideoID":24842,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24842.jpeg","UploadDate":"2021-06-23T01:55:17.7500000","DurationForVideoObject":"PT7M31S","Description":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":25610},{"Watched":false,"Name":"Exercise 5","Duration":"1m 24s","ChapterTopicVideoID":24833,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24833.jpeg","UploadDate":"2021-06-23T01:48:20.6670000","DurationForVideoObject":"PT1M24S","Description":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":25601},{"Watched":false,"Name":"Exercise 6","Duration":"3m 43s","ChapterTopicVideoID":24834,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24834.jpeg","UploadDate":"2021-06-23T01:49:07.4700000","DurationForVideoObject":"PT3M43S","Description":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":25602},{"Watched":false,"Name":"Exercise 7","Duration":"1m 49s","ChapterTopicVideoID":24835,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24835.jpeg","UploadDate":"2021-06-23T01:49:29.6070000","DurationForVideoObject":"PT1M49S","Description":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":25603},{"Watched":false,"Name":"Intervals on the Number Line","Duration":"3m 59s","ChapterTopicVideoID":24839,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24839.jpeg","UploadDate":"2021-06-23T01:51:51.2070000","DurationForVideoObject":"PT3M59S","Description":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":25607},{"Watched":false,"Name":"Exercise 8","Duration":"3m 22s","ChapterTopicVideoID":24836,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24836.jpeg","UploadDate":"2021-06-23T01:50:02.9730000","DurationForVideoObject":"PT3M22S","Description":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":25604},{"Watched":false,"Name":"Exercise 9","Duration":"1m 53s","ChapterTopicVideoID":24837,"CourseChapterTopicPlaylistID":45108,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24837.jpeg","UploadDate":"2021-06-23T01:50:25.0270000","DurationForVideoObject":"PT1M53S","Description":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":25605}],"Thumbnail":null,"ID":45108},{"Name":"Operations on Sets","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Union and Intersection","Duration":"7m 33s","ChapterTopicVideoID":24856,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24856.jpeg","UploadDate":"2021-06-23T04:11:44.7970000","DurationForVideoObject":"PT7M33S","Description":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":25624},{"Watched":false,"Name":"Exercise 1","Duration":"3m 16s","ChapterTopicVideoID":24858,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24858.jpeg","UploadDate":"2021-06-23T04:12:43.1230000","DurationForVideoObject":"PT3M16S","Description":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":25626},{"Watched":false,"Name":"Exercise 2","Duration":"3m 12s","ChapterTopicVideoID":24859,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24859.jpeg","UploadDate":"2021-06-23T04:13:25.0430000","DurationForVideoObject":"PT3M12S","Description":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":25627},{"Watched":false,"Name":"Set difference","Duration":"3m 51s","ChapterTopicVideoID":24854,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24854.jpeg","UploadDate":"2021-06-23T04:09:39.9430000","DurationForVideoObject":"PT3M51S","Description":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":25622},{"Watched":false,"Name":"Exercise 3","Duration":"1m 54s","ChapterTopicVideoID":24860,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24860.jpeg","UploadDate":"2021-06-23T04:13:45.8930000","DurationForVideoObject":"PT1M54S","Description":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":25628},{"Watched":false,"Name":"Exercise 4","Duration":"2m 15s","ChapterTopicVideoID":24861,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24861.jpeg","UploadDate":"2021-06-23T04:14:10.5470000","DurationForVideoObject":"PT2M15S","Description":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":25629},{"Watched":false,"Name":"Complement of a Set","Duration":"4m 35s","ChapterTopicVideoID":24846,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24846.jpeg","UploadDate":"2021-06-23T04:04:30.6970000","DurationForVideoObject":"PT4M35S","Description":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":25614},{"Watched":false,"Name":"Exercise 5","Duration":"2m 30s","ChapterTopicVideoID":24862,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24862.jpeg","UploadDate":"2021-06-23T04:14:40.2100000","DurationForVideoObject":"PT2M30S","Description":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":25630},{"Watched":false,"Name":"Exercise 6","Duration":"3m 58s","ChapterTopicVideoID":24847,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24847.jpeg","UploadDate":"2021-06-23T04:06:39.3870000","DurationForVideoObject":"PT3M58S","Description":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":25615},{"Watched":false,"Name":"Exercise 7","Duration":"2m 36s","ChapterTopicVideoID":24848,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24848.jpeg","UploadDate":"2021-06-23T04:07:14.7970000","DurationForVideoObject":"PT2M36S","Description":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":25616},{"Watched":false,"Name":"Venn Diagrams","Duration":"1m 29s","ChapterTopicVideoID":24857,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24857.jpeg","UploadDate":"2021-06-23T04:12:01.2870000","DurationForVideoObject":"PT1M29S","Description":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":25625},{"Watched":false,"Name":"Exercise 8","Duration":"2m 33s","ChapterTopicVideoID":24849,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24849.jpeg","UploadDate":"2021-06-23T04:07:43.2430000","DurationForVideoObject":"PT2M33S","Description":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":25617},{"Watched":false,"Name":"Exercise 9","Duration":"3m 34s","ChapterTopicVideoID":24850,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24850.jpeg","UploadDate":"2021-06-23T04:08:21.6930000","DurationForVideoObject":"PT3M34S","Description":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":25618},{"Watched":false,"Name":"The Power Set","Duration":"4m 21s","ChapterTopicVideoID":24855,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24855.jpeg","UploadDate":"2021-06-23T04:10:25.4400000","DurationForVideoObject":"PT4M21S","Description":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":25623},{"Watched":false,"Name":"Exercise 10","Duration":"1m 1s","ChapterTopicVideoID":24851,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24851.jpeg","UploadDate":"2021-06-23T04:08:31.7330000","DurationForVideoObject":"PT1M1S","Description":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":25619},{"Watched":false,"Name":"Exercise 11","Duration":"1m 44s","ChapterTopicVideoID":24852,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24852.jpeg","UploadDate":"2021-06-23T04:08:49.0030000","DurationForVideoObject":"PT1M44S","Description":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":25620},{"Watched":false,"Name":"Exercise 12","Duration":"1m 14s","ChapterTopicVideoID":24853,"CourseChapterTopicPlaylistID":45109,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24853.jpeg","UploadDate":"2021-06-23T04:09:00.4770000","DurationForVideoObject":"PT1M14S","Description":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":25621}],"Thumbnail":null,"ID":45109},{"Name":"Irrational Numbers","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Preface","Duration":"1m 41s","ChapterTopicVideoID":24867,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24867.jpeg","UploadDate":"2021-06-23T04:26:37.8300000","DurationForVideoObject":"PT1M41S","Description":null,"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":25635},{"Watched":false,"Name":"Reduced Fractions","Duration":"2m 7s","ChapterTopicVideoID":24863,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24863.jpeg","UploadDate":"2021-06-23T04:20:33.2230000","DurationForVideoObject":"PT2M7S","Description":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":25631},{"Watched":false,"Name":"Fundamental Theorem of Arithmetic","Duration":"5m 35s","ChapterTopicVideoID":24866,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24866.jpeg","UploadDate":"2021-06-23T04:26:18.3300000","DurationForVideoObject":"PT5M35S","Description":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":25634},{"Watched":false,"Name":"Exercise 1","Duration":"3m 30s","ChapterTopicVideoID":24868,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24868.jpeg","UploadDate":"2021-06-23T04:27:21.7400000","DurationForVideoObject":"PT3M30S","Description":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":25636},{"Watched":false,"Name":"Exercise 2","Duration":"4m 45s","ChapterTopicVideoID":24869,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24869.jpeg","UploadDate":"2021-06-23T04:28:15.2670000","DurationForVideoObject":"PT4M45S","Description":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":25637},{"Watched":false,"Name":"Exercise 3","Duration":"3m 35s","ChapterTopicVideoID":24870,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24870.jpeg","UploadDate":"2021-06-23T04:28:55.5000000","DurationForVideoObject":"PT3M35S","Description":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":25638},{"Watched":false,"Name":"Exercise 4","Duration":"3m 15s","ChapterTopicVideoID":24871,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24871.jpeg","UploadDate":"2021-06-23T04:29:30.1130000","DurationForVideoObject":"PT3M15S","Description":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":25639},{"Watched":false,"Name":"Exercise 5","Duration":"3m 45s","ChapterTopicVideoID":24864,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24864.jpeg","UploadDate":"2021-06-23T04:23:59.6270000","DurationForVideoObject":"PT3M45S","Description":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":25632},{"Watched":false,"Name":"Exercise 6","Duration":"5m 30s","ChapterTopicVideoID":24865,"CourseChapterTopicPlaylistID":45110,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/24865.jpeg","UploadDate":"2021-06-23T04:25:04.0470000","DurationForVideoObject":"PT5M30S","Description":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":25633}],"Thumbnail":null,"ID":45110}]

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