1 00:00:00,680 --> 00:00:04,170 Probability models often involve infinite sample 2 00:00:04,170 --> 00:00:08,780 spaces, that is, infinite sets. 3 00:00:08,780 --> 00:00:12,570 But not all sets are of the same kind. 4 00:00:12,570 --> 00:00:17,530 Some sets are discrete and we call them countable, and some 5 00:00:17,530 --> 00:00:20,960 are continuous and we call them uncountable. 6 00:00:20,960 --> 00:00:23,470 But what exactly is the difference between these two 7 00:00:23,470 --> 00:00:24,860 types of sets? 8 00:00:24,860 --> 00:00:27,760 How can we define it precisely? 9 00:00:27,760 --> 00:00:31,320 Well, let us start by first giving a definition of what it 10 00:00:31,320 --> 00:00:33,920 means to have a countable set. 11 00:00:33,920 --> 00:00:37,530 A set will be called countable if its elements can be put 12 00:00:37,530 --> 00:00:41,450 into a 1-to-1 correspondence with the positive integers. 13 00:00:41,450 --> 00:00:45,280 This means that we look at the elements of that set, and we 14 00:00:45,280 --> 00:00:47,990 take one element-- we call it the first element. 15 00:00:47,990 --> 00:00:50,790 We take another element-- we call it the second. 16 00:00:50,790 --> 00:00:54,090 Another, we call the third element, and so on. 17 00:00:54,090 --> 00:00:58,980 And this way we will eventually exhaust all of the 18 00:00:58,980 --> 00:01:02,920 elements of the set, so that each one of those elements 19 00:01:02,920 --> 00:01:06,840 corresponds to a particular positive integer, namely the 20 00:01:06,840 --> 00:01:09,140 index that appears underneath. 21 00:01:09,140 --> 00:01:13,110 More formally, what's happening is that we take 22 00:01:13,110 --> 00:01:18,810 elements of that set that are arranged in a sequence. 23 00:01:18,810 --> 00:01:24,420 We look at the set, which is the entire range of values of 24 00:01:24,420 --> 00:01:28,580 that sequence, and we want that sequence to exhaust the 25 00:01:28,580 --> 00:01:31,470 entire set omega. 26 00:01:31,470 --> 00:01:37,250 Or in other words, in simpler terms, we want to be able to 27 00:01:37,250 --> 00:01:43,360 arrange all of the elements of omega in a sequence. 28 00:01:43,360 --> 00:01:46,800 So what are some examples of countable sets? 29 00:01:46,800 --> 00:01:50,330 In a trivial sense, the positive integers themselves 30 00:01:50,330 --> 00:01:54,960 are countable, because we can arrange them in a sequence. 31 00:01:54,960 --> 00:01:58,110 This is almost tautological, by the definition. 32 00:01:58,110 --> 00:02:00,820 For a more interesting example, let's look at the set 33 00:02:00,820 --> 00:02:02,370 of all integers. 34 00:02:02,370 --> 00:02:04,610 Can we arrange them in a sequence? 35 00:02:04,610 --> 00:02:09,669 Yes, we can, and we can do it in this manner, where we 36 00:02:09,669 --> 00:02:13,330 alternate between positive and negative numbers. 37 00:02:13,330 --> 00:02:16,880 And this way, we're going to cover all of the integers, and 38 00:02:16,880 --> 00:02:20,190 we have arranged them in a sequence. 39 00:02:20,190 --> 00:02:24,870 How about the set of all pairs of positive integers? 40 00:02:24,870 --> 00:02:27,280 This is less clear. 41 00:02:27,280 --> 00:02:28,480 Let us look at this picture. 42 00:02:28,480 --> 00:02:32,560 This is the set of all pairs of positive integers, which we 43 00:02:32,560 --> 00:02:35,590 understand to continue indefinitely. 44 00:02:35,590 --> 00:02:39,450 Can we arrange this sets in a sequence? 45 00:02:39,450 --> 00:02:40,870 It turns out that we can. 46 00:02:40,870 --> 00:02:45,310 And we can do it by tracing a path of this kind. 47 00:02:50,130 --> 00:02:52,990 So you can probably get the sense of how 48 00:02:52,990 --> 00:02:54,820 this path is going. 49 00:02:54,820 --> 00:03:00,090 And by continuing this way, over and over, we're going to 50 00:03:00,090 --> 00:03:05,370 cover the entire set of all pairs of positive integers. 51 00:03:05,370 --> 00:03:07,840 So we have managed to arrange them in a sequence. 52 00:03:07,840 --> 00:03:11,870 So the set of all such pairs is indeed a countable set. 53 00:03:11,870 --> 00:03:15,160 And the same argument can be extended to argue for the set 54 00:03:15,160 --> 00:03:19,816 of all triples of positive integers, or the set of all 55 00:03:19,816 --> 00:03:23,350 quadruples of positive integers, and so on. 56 00:03:23,350 --> 00:03:28,550 This is actually not just a trivial mathematical point 57 00:03:28,550 --> 00:03:32,270 that we discuss for some curious reason, but it is 58 00:03:32,270 --> 00:03:35,280 because we will often have sample spaces 59 00:03:35,280 --> 00:03:36,960 that are of this kind. 60 00:03:36,960 --> 00:03:41,490 And it's important to know that they're countable. 61 00:03:41,490 --> 00:03:44,079 Now for a more subtle example. 62 00:03:44,079 --> 00:03:48,070 Let us look at all rational numbers within the range 63 00:03:48,070 --> 00:03:50,640 between 0 and 1. 64 00:03:50,640 --> 00:03:53,140 What do we mean by rational numbers? 65 00:03:53,140 --> 00:03:56,340 We mean those numbers that can be expressed as a 66 00:03:56,340 --> 00:03:59,050 ratio of two integers. 67 00:03:59,050 --> 00:04:02,240 It turns out that we can arrange them in a sequence, 68 00:04:02,240 --> 00:04:04,540 and we can do it as follows. 69 00:04:04,540 --> 00:04:07,050 Let us first look at rational numbers that have a 70 00:04:07,050 --> 00:04:09,450 denominator term of 2. 71 00:04:09,450 --> 00:04:12,120 Then, look at the rational numbers that have a 72 00:04:12,120 --> 00:04:16,140 denominator term of 3. 73 00:04:16,140 --> 00:04:19,500 Then, look at the rational numbers, always within this 74 00:04:19,500 --> 00:04:23,230 range of interest, that have a denominator of 4. 75 00:04:26,270 --> 00:04:28,780 And then we continue similarly-- 76 00:04:28,780 --> 00:04:33,630 rational numbers that have a denominator of 5, and so on. 77 00:04:33,630 --> 00:04:35,909 This way, we're going to exhaust all of 78 00:04:35,909 --> 00:04:37,710 the rational numbers. 79 00:04:37,710 --> 00:04:42,650 Actually, this number here already appeared there. 80 00:04:42,650 --> 00:04:44,030 It's the same number. 81 00:04:44,030 --> 00:04:47,720 So we do not need to include this in a sequence, but that's 82 00:04:47,720 --> 00:04:49,020 not an issue. 83 00:04:49,020 --> 00:04:52,920 Whenever we see a rational number that has already been 84 00:04:52,920 --> 00:04:56,060 encountered before, we just delete it. 85 00:04:56,060 --> 00:05:00,480 In the end, we end up with a sequence that goes over all of 86 00:05:00,480 --> 00:05:02,090 the possible rational numbers. 87 00:05:02,090 --> 00:05:05,310 And so we conclude that the set of all rational numbers is 88 00:05:05,310 --> 00:05:07,500 itself a countable set. 89 00:05:07,500 --> 00:05:11,350 So what kind of set would be uncountable? 90 00:05:11,350 --> 00:05:14,240 An uncountable set, by definition, is a set that is 91 00:05:14,240 --> 00:05:15,570 not countable. 92 00:05:15,570 --> 00:05:19,480 And there are examples of uncountable sets, most 93 00:05:19,480 --> 00:05:24,560 prominent, continuous subsets of the real line. 94 00:05:24,560 --> 00:05:28,110 Whenever we have an interval, the unit interval, or any 95 00:05:28,110 --> 00:05:31,710 other interval that has positive length, that interval 96 00:05:31,710 --> 00:05:34,290 is an uncountable set. 97 00:05:34,290 --> 00:05:37,370 And the same is true if, instead of an interval, we 98 00:05:37,370 --> 00:05:40,490 look at the entire real line, or we look at the 99 00:05:40,490 --> 00:05:43,330 two-dimensional plane, or three-dimensional 100 00:05:43,330 --> 00:05:44,760 space, and so on. 101 00:05:44,760 --> 00:05:48,960 So all the usual sets that we think of as continuous sets 102 00:05:48,960 --> 00:05:52,040 turn out to be uncountable. 103 00:05:52,040 --> 00:05:55,040 How do we know that they are uncountable? 104 00:05:55,040 --> 00:05:58,640 There is actually a brilliant argument that establishes that 105 00:05:58,640 --> 00:06:01,650 the unit interval is uncountable. 106 00:06:01,650 --> 00:06:04,360 And then the argument is easily extended to other 107 00:06:04,360 --> 00:06:07,150 cases, like the reals and the plane. 108 00:06:07,150 --> 00:06:10,240 We do not need to know how this argument goes, for the 109 00:06:10,240 --> 00:06:11,820 purposes of this course. 110 00:06:11,820 --> 00:06:15,930 But just because it is so beautiful, we will actually be 111 00:06:15,930 --> 00:06:17,190 presenting it to you.