WEBVTT 00:00:00.250 --> 00:00:03.722 A truth table gives a formula a true or false value at 00:00:03.722 --> 00:00:05.108 each complete assignment. 00:00:05.595 --> 00:00:08.646 Today we collect all the true rows into a set. 00:00:09.550 --> 00:00:11.264 Keep our four states fixed. 00:00:11.801 --> 00:00:16.238 A has a dot only, B has both features, C has a 00:00:16.238 --> 00:00:18.883 stripe only, and D has neither. 00:00:19.498 --> 00:00:22.839 X means dot, and Y means stripe. 00:00:23.755 --> 00:00:25.610 X is true at A and B. 00:00:26.444 --> 00:00:30.198 Its truth region is therefore the set containing A and B. 00:00:31.007 --> 00:00:35.565 The paper's symbol on screen names this truth-region operation. 00:00:36.694 --> 00:00:41.294 Keep three things separate: X is a formula; true is its value 00:00:41.294 --> 00:00:45.104 at A; and the set containing A and B is its truth 00:00:45.135 --> 00:00:45.690 region. 00:00:46.314 --> 00:00:51.273 We are describing every possible state, not announcing which hidden card we 00:00:51.295 --> 00:00:52.025 have. 00:00:52.913 --> 00:00:54.509 X is true at A and B. 00:00:55.361 --> 00:00:57.250 Y is true at B and C. 00:00:58.059 --> 00:01:02.287 For X and Y, we need both requirements in the same state. 00:01:02.674 --> 00:01:05.702 Which state belongs to both input regions? 00:01:15.983 --> 00:01:16.602 Only B. 00:01:17.441 --> 00:01:19.925 That shared part is the intersection. 00:01:20.396 --> 00:01:24.030 So the truth region of X and Y is the singleton containing 00:01:24.042 --> 00:01:24.226 B. 00:01:25.480 --> 00:01:30.503 The rule works for any two formulas: a state satisfies their AND 00:01:30.503 --> 00:01:33.727 exactly when it belongs to both truth regions. 00:01:34.346 --> 00:01:37.193 That is why AND becomes intersection. 00:01:38.093 --> 00:01:40.990 For X or Y, at least one requirement is enough. 00:01:41.624 --> 00:01:43.165 A qualifies through X. 00:01:43.825 --> 00:01:45.389 B qualifies through both. 00:01:46.020 --> 00:01:47.627 C qualifies through Y. 00:01:48.231 --> 00:01:50.222 D qualifies through neither. 00:01:51.088 --> 00:01:53.877 The resulting region contains A, B and C. 00:01:54.496 --> 00:01:57.035 This is the union of the two input regions. 00:01:57.639 --> 00:02:01.286 B is included once, even though both inputs include it. 00:02:02.209 --> 00:02:04.726 Inclusive OR keeps the overlap. 00:02:05.196 --> 00:02:10.364 We will soon compare it with exclusive-or, which removes that overlap. 00:02:11.279 --> 00:02:14.597 Not X is true exactly where X is false. 00:02:15.058 --> 00:02:17.499 In this universe, that is C and D. 00:02:18.313 --> 00:02:22.469 Its truth region is the complement of the X region inside our 00:02:22.480 --> 00:02:23.557 fixed carrier. 00:02:24.510 --> 00:02:27.429 Likewise, not Y is true at A and D. 00:02:28.254 --> 00:02:32.857 C has a stripe, so C does not satisfy not Y. 00:02:33.972 --> 00:02:36.277 Intersect X with not Y. 00:02:36.747 --> 00:02:41.543 A is in both regions; B fails not Y; C and D 00:02:41.694 --> 00:02:42.552 fail X. 00:02:43.248 --> 00:02:46.688 So X and not Y describes A alone. 00:02:47.813 --> 00:02:52.729 Complement always means outside the chosen set but still inside the stated 00:02:52.739 --> 00:02:53.684 universe. 00:02:54.120 --> 00:02:59.375 Describing the no-dot region does not, by itself, give an observer 00:02:59.396 --> 00:03:01.088 a no-dot certificate. 00:03:01.993 --> 00:03:07.176 Exclusive-or, or XOR, is true when the two inputs differ. 00:03:07.800 --> 00:03:11.727 The double vertical arrow is our written symbol for it. 00:03:12.459 --> 00:03:16.940 At A, X is one and Y is zero: they differ, so 00:03:17.006 --> 00:03:18.590 XOR is one. 00:03:19.018 --> 00:03:23.358 At B, both are one: they agree, so XOR is zero. 00:03:24.483 --> 00:03:28.581 At C, zero and one differ, so XOR is one. 00:03:29.024 --> 00:03:32.855 At D, both are zero, so XOR is zero. 00:03:33.469 --> 00:03:35.649 We have now checked every row. 00:03:36.504 --> 00:03:39.542 The XOR truth region contains A and C. 00:03:40.371 --> 00:03:43.355 Compare it with OR, which also contains B. 00:03:44.164 --> 00:03:49.540 OR accepts one or both features; XOR accepts exactly one. 00:03:50.390 --> 00:03:55.768 The union splits into two disjoint pieces: the overlap, B, and the 00:03:55.790 --> 00:03:57.989 exclusive remainder, A and C. 00:03:58.733 --> 00:04:01.818 Disjoint means the pieces share no state. 00:04:02.405 --> 00:04:05.748 For any two sets, the part in one but not both is 00:04:05.758 --> 00:04:07.757 called their symmetric difference. 00:04:08.192 --> 00:04:11.826 That is the set operation corresponding to XOR. 00:04:12.297 --> 00:04:17.007 Its triangle symbol here means symmetric difference, not a boundary. 00:04:17.757 --> 00:04:22.174 Two formulas are semantically equivalent when they have the same truth value 00:04:22.174 --> 00:04:23.552 at every valuation. 00:04:24.671 --> 00:04:27.101 Now compare X with X or Y. 00:04:27.655 --> 00:04:29.875 Their truth regions overlap. 00:04:30.175 --> 00:04:32.259 Does that make the formulas equivalent? 00:04:32.722 --> 00:04:35.811 If not, name a state where their values differ. 00:04:50.050 --> 00:04:50.397 No. 00:04:51.031 --> 00:04:54.675 At C, X is false while X or Y is true. 00:04:55.220 --> 00:04:57.196 Overlap is not equality. 00:04:57.773 --> 00:05:03.155 C is a counterexample to equivalence, and the symmetric difference here is 00:05:03.176 --> 00:05:05.042 the singleton containing C. 00:05:06.257 --> 00:05:11.446 Compare X or Y with this second expression: NOT, applied to the 00:05:11.446 --> 00:05:14.903 whole conjunction NOT X AND NOT Y. 00:05:15.420 --> 00:05:17.333 Work from the inside outward. 00:05:17.789 --> 00:05:23.659 First evaluate the two inner negations and their AND; only then reverse 00:05:23.659 --> 00:05:26.381 that AND result with the outer NOT. 00:05:26.906 --> 00:05:29.029 The expressions are written differently. 00:05:29.596 --> 00:05:32.494 Do they describe the same Boolean rule? 00:05:33.203 --> 00:05:37.761 At A, not X is false, so the inner AND is false 00:05:37.761 --> 00:05:39.526 and its negation is true. 00:05:39.945 --> 00:05:42.995 At B the inner AND is again false. 00:05:44.279 --> 00:05:49.246 At C, not Y is false, so the result is true again. 00:05:50.351 --> 00:05:53.506 At D, both inner negations are true. 00:05:53.915 --> 00:05:57.488 Their AND is true, so its negation is false. 00:05:57.954 --> 00:06:03.455 Both formulas therefore have outputs one, one, one, zero in our row 00:06:03.467 --> 00:06:03.890 order. 00:06:05.628 --> 00:06:11.364 Their truth regions are equal, so the formulas are semantically equivalent: they 00:06:11.364 --> 00:06:15.043 agree at every valuation in the fixed universe. 00:06:15.479 --> 00:06:18.459 Their written expressions are still different. 00:06:19.258 --> 00:06:20.929 Why is this check enough? 00:06:21.574 --> 00:06:25.236 Our table contains every assignment of X and Y. 00:06:25.705 --> 00:06:28.989 One differing row would disprove equivalence. 00:06:29.311 --> 00:06:34.150 Complete agreement over all four rows proves it for these two-input 00:06:34.161 --> 00:06:34.942 formulas. 00:06:36.073 --> 00:06:42.636 More generally, equal truth regions mean no valuation makes exactly one formula 00:06:42.646 --> 00:06:43.196 true. 00:06:43.615 --> 00:06:45.794 Their XOR region is empty. 00:06:46.255 --> 00:06:50.515 A nonempty XOR region supplies a counterexample. 00:06:51.283 --> 00:06:52.757 Try a new comparison. 00:06:53.218 --> 00:06:58.271 Is Y and not X equivalent to X exclusive-or Y? 00:06:58.763 --> 00:07:02.941 Find their truth regions and a counterexample if they differ. 00:07:21.212 --> 00:07:22.018 They differ. 00:07:22.531 --> 00:07:25.135 Y and not X is true only at C. 00:07:25.928 --> 00:07:27.892 XOR is true at A and C. 00:07:28.656 --> 00:07:33.300 A is a counterexample: XOR is true there, but Y and not 00:07:33.300 --> 00:07:34.993 X is false. 00:07:35.583 --> 00:07:41.296 We can now translate AND into intersection, OR into union, NOT into 00:07:41.317 --> 00:07:45.095 complement, and XOR into symmetric difference. 00:07:45.505 --> 00:07:49.717 Equal truth regions characterize semantic equivalence. 00:07:50.662 --> 00:07:56.532 Next, complete descriptions will show how individual states build every finite truth 00:07:56.554 --> 00:07:57.108 region. 00:07:57.772 --> 00:08:02.916 We will then develop distance and topology before choosing which truth regions 00:08:02.948 --> 00:08:05.390 count as available observations.