1 00:00:01,230 --> 00:00:03,780 We will now discuss De Morgan's laws that are some 2 00:00:03,780 --> 00:00:08,109 very useful relations between sets and their complements. 3 00:00:08,109 --> 00:00:11,500 One of the De Morgan's laws takes this form. 4 00:00:11,500 --> 00:00:14,360 If we take the intersection of two sets and then take the 5 00:00:14,360 --> 00:00:17,500 complement of this intersection, what we obtain 6 00:00:17,500 --> 00:00:21,070 is the union of the complements of the two sets. 7 00:00:21,070 --> 00:00:23,840 Pictorially, here is the situation. 8 00:00:23,840 --> 00:00:26,680 We have our universal set. 9 00:00:26,680 --> 00:00:34,210 Inside that set, we have a set, S, which is this one. 10 00:00:34,210 --> 00:00:40,340 And we have another set, T, which is this one. 11 00:00:40,340 --> 00:00:42,840 Let us look at this side. 12 00:00:42,840 --> 00:00:47,950 The complement of S is this part of the diagram. 13 00:00:47,950 --> 00:00:52,950 The complement of T is this part of the diagram. 14 00:00:52,950 --> 00:00:54,450 What is left? 15 00:00:54,450 --> 00:00:57,680 What is left is just this region here, which is the 16 00:00:57,680 --> 00:01:03,740 intersection of S with T. So anything that does not belong 17 00:01:03,740 --> 00:01:07,010 here belongs to the intersection. 18 00:01:07,010 --> 00:01:10,950 This means that the complement of the intersection is 19 00:01:10,950 --> 00:01:14,570 everything out there, which is the set. 20 00:01:14,570 --> 00:01:18,180 If you're not convinced by this pictorial proof, let us 21 00:01:18,180 --> 00:01:22,470 go through an argument that is a little more formal. 22 00:01:22,470 --> 00:01:26,360 What does it take for an element to belong 23 00:01:26,360 --> 00:01:29,870 to the first set? 24 00:01:29,870 --> 00:01:32,729 In order to belong to that set, x belongs to the 25 00:01:32,729 --> 00:01:37,539 complement of S intersection T. This is the same as saying 26 00:01:37,539 --> 00:01:42,020 that x does not belong to the intersection [of] 27 00:01:42,020 --> 00:01:44,450 S with T. 28 00:01:44,450 --> 00:01:46,370 What does that mean? 29 00:01:46,370 --> 00:01:49,500 Since it is not in the intersection, this is the same 30 00:01:49,500 --> 00:01:56,940 as saying that x does not belong to S or x does not 31 00:01:56,940 --> 00:02:04,640 belong to T. But this is the same as saying that x belongs 32 00:02:04,640 --> 00:02:10,580 to the complement of S or x belongs to the complement of 33 00:02:10,580 --> 00:02:17,650 T. And this is equivalent to saying that x belongs to the 34 00:02:17,650 --> 00:02:23,310 union of the complement of S with the complement of T. 35 00:02:23,310 --> 00:02:26,610 So this establishes this first De Morgan's law. 36 00:02:29,750 --> 00:02:33,350 There's another De Morgan's law, which is obtained from 37 00:02:33,350 --> 00:02:36,110 this one by a syntactic substitution. 38 00:02:36,110 --> 00:02:38,450 We're going to play the following trick. 39 00:02:38,450 --> 00:02:42,520 Wherever we see an S, we're going to replace it by S 40 00:02:42,520 --> 00:02:43,410 complement. 41 00:02:43,410 --> 00:02:47,290 And wherever we see an S complement, we will replace it 42 00:02:47,290 --> 00:02:48,620 with an S. 43 00:02:48,620 --> 00:02:52,440 And similarly, whenever we see a T, we'll replace it by T 44 00:02:52,440 --> 00:02:53,150 complement. 45 00:02:53,150 --> 00:02:56,470 And when we see a T complement, we will replace it 46 00:02:56,470 --> 00:03:01,770 by T. So doing this syntactic substitution, what we obtain 47 00:03:01,770 --> 00:03:06,100 is S complement intersection with T complement-- 48 00:03:06,100 --> 00:03:08,280 everything gets complemented-- 49 00:03:08,280 --> 00:03:14,200 is the same as S union T. 50 00:03:14,200 --> 00:03:16,560 Now, let us take complements of both sides. 51 00:03:16,560 --> 00:03:19,760 The complement of a complement is the set itself. 52 00:03:19,760 --> 00:03:22,690 So we obtain this. 53 00:03:22,690 --> 00:03:25,880 And now, we take the complement of the other side, 54 00:03:25,880 --> 00:03:27,970 which is this one. 55 00:03:27,970 --> 00:03:31,640 And this is the second De Morgan's law. 56 00:03:31,640 --> 00:03:35,930 It tells us that the complement of a union is the 57 00:03:35,930 --> 00:03:39,270 same as the intersection of the complements. 58 00:03:39,270 --> 00:03:43,010 We derived it from the first De Morgan's law by a syntactic 59 00:03:43,010 --> 00:03:43,980 substitution. 60 00:03:43,980 --> 00:03:47,630 If you're not convinced, it would be useful for you to go 61 00:03:47,630 --> 00:03:52,040 through an argument of this kind to show that something is 62 00:03:52,040 --> 00:03:55,400 an element of this set if and only if it is an element of 63 00:03:55,400 --> 00:03:57,870 that set as well. 64 00:03:57,870 --> 00:04:02,000 Finally, it turns out that De Morgan's laws are valid when 65 00:04:02,000 --> 00:04:04,530 we take unions or intersections of 66 00:04:04,530 --> 00:04:06,480 more than two sets. 67 00:04:06,480 --> 00:04:08,390 There is a more general form. 68 00:04:08,390 --> 00:04:11,390 And the general form is as follows-- 69 00:04:11,390 --> 00:04:13,370 an analogy with this one. 70 00:04:13,370 --> 00:04:17,089 If we have a collection of sets, Sn, perhaps an infinite 71 00:04:17,089 --> 00:04:20,160 collection, we take the intersection of those sets and 72 00:04:20,160 --> 00:04:24,580 then the complement, what that is is the union of the 73 00:04:24,580 --> 00:04:26,350 complements. 74 00:04:26,350 --> 00:04:28,540 So this is analygous to this law. 75 00:04:28,540 --> 00:04:32,840 And this law extends to this one: if we have the union of 76 00:04:32,840 --> 00:04:36,455 certain sets and we take the complement of the union, what 77 00:04:36,455 --> 00:04:41,220 we obtain is the intersection of the complements. 78 00:04:41,220 --> 00:04:44,409 We will have many occasions to use De Morgan's laws. 79 00:04:44,409 --> 00:04:46,370 They're actually very useful. 80 00:04:46,370 --> 00:04:50,230 They allow us, in general, to go back and forth between 81 00:04:50,230 --> 00:04:51,560 unions and intersections.