WEBVTT 00:00:00.250 --> 00:00:03.818 Our cards use X for a dot and Y for a stripe. 00:00:04.279 --> 00:00:07.566 A has only a dot; B has both; C has only a 00:00:07.566 --> 00:00:09.112 stripe; D has neither. 00:00:09.664 --> 00:00:14.017 In each two-bit annotation, X comes first and Y second. 00:00:14.500 --> 00:00:17.937 One means true, and zero means false. 00:00:18.465 --> 00:00:21.828 A formula’s truth region is the set of cards where it 00:00:21.828 --> 00:00:22.592 is true. 00:00:23.590 --> 00:00:27.300 Imagine sorting all four state cards into boxes. 00:00:27.895 --> 00:00:32.352 Every card must enter a box, no card may enter two boxes, 00:00:32.591 --> 00:00:34.161 and no box is empty. 00:00:34.743 --> 00:00:39.165 Such a collection of boxes is called a partition of the carrier. 00:00:40.010 --> 00:00:43.187 The dot states A and B make one block. 00:00:43.648 --> 00:00:46.964 The no-dot states C and D make the other. 00:00:47.454 --> 00:00:51.952 Their union is the whole carrier, and their intersection is empty. 00:00:52.413 --> 00:00:56.873 This is one partition, with two blocks rather than four points. 00:00:57.982 --> 00:01:03.122 Now describe A completely: X is true and Y is false. 00:01:03.508 --> 00:01:07.582 The formula X and not Y selects A alone. 00:01:08.025 --> 00:01:12.821 A conjunction that specifies a truth value for every variable is called 00:01:12.821 --> 00:01:14.219 a complete minterm. 00:01:15.278 --> 00:01:18.086 B requires both variables true. 00:01:18.535 --> 00:01:21.236 C requires not X and Y. 00:01:21.856 --> 00:01:24.759 D requires not X and not Y. 00:01:25.348 --> 00:01:30.925 These descriptions are mutually exclusive: no state satisfies two of them. 00:01:30.957 --> 00:01:34.592 They are exhaustive: every state satisfies one. 00:01:35.197 --> 00:01:39.851 Together, they describe every complete assignment exactly once. 00:01:40.765 --> 00:01:44.325 Suppose our target is the set containing A and C. 00:01:45.061 --> 00:01:47.549 Start with the complete description of A. 00:01:48.109 --> 00:01:53.781 It contributes the singleton containing A, rather than the entire region where 00:01:53.781 --> 00:01:55.088 X is true. 00:01:55.988 --> 00:02:00.078 The complete description of C contributes the other singleton. 00:02:00.708 --> 00:02:05.655 A state belongs to the target when it satisfies either description. 00:02:06.285 --> 00:02:15.171 The word either here is inclusive OR; disjointness makes simultaneous satisfaction impossible 00:02:15.171 --> 00:02:15.818 anyway. 00:02:16.806 --> 00:02:20.560 The disjunction therefore defines exactly A and C. 00:02:21.727 --> 00:02:26.084 XOR means exactly one of the two inputs is true. 00:02:26.896 --> 00:02:28.741 This is our XOR region. 00:02:29.345 --> 00:02:34.184 We built it from complete descriptions instead of guessing a shorter formula 00:02:34.217 --> 00:02:36.852 and hoping its truth region matched. 00:02:37.752 --> 00:02:42.426 The power set means the collection of every subset of the carrier. 00:02:42.941 --> 00:02:48.422 Each of four points can independently be included or omitted, giving sixteen 00:02:48.454 --> 00:02:49.228 subsets. 00:02:49.809 --> 00:02:54.704 A point, a subset, and this collection of subsets are different kinds 00:02:54.704 --> 00:02:55.637 of object. 00:02:56.582 --> 00:03:01.443 To describe any target subset in this finite-variable setting, take the 00:03:01.443 --> 00:03:06.706 complete minterm of each point it contains and join those minterms by 00:03:06.760 --> 00:03:07.354 OR. 00:03:07.838 --> 00:03:12.373 A point satisfies that formula exactly when it is one of the 00:03:12.373 --> 00:03:13.438 chosen points. 00:03:14.036 --> 00:03:18.507 That explains why the construction works for every subset of our finite 00:03:18.507 --> 00:03:19.976 valuation carrier. 00:03:20.902 --> 00:03:25.032 For the empty target there are no chosen minterms, so the empty 00:03:25.054 --> 00:03:26.965 disjunction means false. 00:03:27.401 --> 00:03:31.673 An empty conjunction imposes no requirements and means true. 00:03:32.134 --> 00:03:37.061 These conventions keep the same construction valid at the extreme cases. 00:03:38.211 --> 00:03:43.506 Two formulas are semantically equivalent when they agree at every valuation. 00:03:44.120 --> 00:03:49.315 For this two-variable carrier, checking all four rows is exhaustive. 00:03:49.734 --> 00:03:55.114 Checking A alone proves only agreement at A, not equivalence over the 00:03:55.114 --> 00:03:55.868 carrier. 00:03:56.760 --> 00:04:02.674 The familiar union, intersection and complement operations on the power set form 00:04:02.685 --> 00:04:04.987 a finite Boolean algebra. 00:04:05.520 --> 00:04:11.260 Here that name packages operations we already understand; it does not add 00:04:11.260 --> 00:04:13.341 a new rule of inference. 00:04:14.140 --> 00:04:18.957 Complete minterms are the atomic pieces of classical truth sets. 00:04:19.592 --> 00:04:25.837 A definable singleton, written as the set containing v, is one valuation 00:04:25.859 --> 00:04:27.377 named by a formula. 00:04:27.847 --> 00:04:33.588 This does not automatically make singletons a basis for an observation topology. 00:04:34.217 --> 00:04:41.179 If the available observations cannot certify that singleton, logical definability alone does 00:04:41.190 --> 00:04:42.340 not make it open. 00:04:43.484 --> 00:04:48.139 Reconstruct the set containing B and D from complete minterms. 00:04:48.829 --> 00:04:53.689 State the two conjunctions and explain why the other two valuations are 00:04:53.689 --> 00:04:54.446 excluded. 00:04:54.967 --> 00:04:57.355 Pause before revealing the formula. 00:05:15.594 --> 00:05:17.685 B contributes X and Y. 00:05:18.213 --> 00:05:20.898 D contributes not X and not Y. 00:05:21.368 --> 00:05:24.135 Their disjunction is true at B and D. 00:05:24.879 --> 00:05:29.751 At A and C the variables disagree, so neither complete conjunction is 00:05:29.761 --> 00:05:30.587 true. 00:05:31.339 --> 00:05:35.300 We can now move in both directions: from a formula to its 00:05:35.300 --> 00:05:39.403 truth region, and from any finite truth region to a formula. 00:05:39.958 --> 00:05:45.583 Later, observations will select which of those definable events can be certified.