WEBVTT 00:00:00.250 --> 00:00:05.228 Interior asks whether the whole smallest open neighborhood fits inside an event. 00:00:05.767 --> 00:00:09.285 Closure asks whether that neighborhood meets the event at all. 00:00:09.720 --> 00:00:14.160 We are using the original four-state carrier and only positive X 00:00:14.170 --> 00:00:14.815 and Y. 00:00:15.291 --> 00:00:20.833 The fixed X,Y values are A: one-zero, B: one-one, 00:00:21.023 --> 00:00:24.714 C: zero-one and D: zero-zero. 00:00:25.277 --> 00:00:28.901 Our first event is classical X implies Y. 00:00:29.423 --> 00:00:32.694 One means true and zero means false. 00:00:33.506 --> 00:00:38.162 The topological boundary is the closure of an event with its interior 00:00:38.194 --> 00:00:38.904 removed. 00:00:39.418 --> 00:00:45.961 It can be understood as observational ambiguity: neighborhoods cannot settle membership wholly 00:00:45.982 --> 00:00:48.342 inside or wholly outside. 00:00:48.730 --> 00:00:52.115 It is not defined by the edge of a shaded shape on 00:00:52.115 --> 00:00:52.849 the screen. 00:00:53.936 --> 00:00:57.572 Retain the implication calculation from Lecture 18. 00:00:58.276 --> 00:01:02.030 Its interior is B and C and its closure is the whole 00:01:02.041 --> 00:01:02.700 carrier. 00:01:03.226 --> 00:01:07.437 Removing the interior from the closure leaves A and D as the 00:01:07.437 --> 00:01:08.170 boundary. 00:01:09.143 --> 00:01:12.601 A is outside the implication truth region. 00:01:13.085 --> 00:01:18.458 Its smallest neighborhood contains A outside and B inside. 00:01:18.893 --> 00:01:23.603 It cannot settle the event in either direction, so a boundary point 00:01:23.657 --> 00:01:25.677 need not satisfy the event. 00:01:27.130 --> 00:01:34.510 D satisfies implication, yet its neighborhood contains both satisfying and falsifying valuations. 00:01:35.154 --> 00:01:37.335 It also lies on the boundary. 00:01:38.324 --> 00:01:42.816 Truth at the centre and ambiguity of its available neighborhood are different 00:01:42.816 --> 00:01:43.576 properties. 00:01:44.705 --> 00:01:49.016 For the next event, XOR means exactly one of X and Y 00:01:49.026 --> 00:01:49.792 is true. 00:01:50.299 --> 00:01:54.479 The carrier and the positive observation menu stay fixed. 00:01:55.469 --> 00:01:58.996 The XOR truth region contains A and C. 00:01:59.780 --> 00:02:04.261 Under positive X and Y, A's and C's smallest neighborhoods 00:02:04.283 --> 00:02:07.753 both include B, which is outside this event. 00:02:08.395 --> 00:02:11.967 No nonempty open set fits inside XOR. 00:02:13.065 --> 00:02:14.755 The interior is empty. 00:02:15.271 --> 00:02:20.359 A, C and D have neighborhoods meeting XOR, while B's singleton 00:02:20.359 --> 00:02:21.717 neighborhood does not. 00:02:22.274 --> 00:02:26.413 The closure, and hence the boundary in this case, is A, C 00:02:26.533 --> 00:02:26.924 and D. 00:02:28.253 --> 00:02:32.615 Make both signs of X and Y available on the same carrier. 00:02:33.117 --> 00:02:38.951 Each complete minterm now certifies a singleton, so the topology is discrete. 00:02:39.293 --> 00:02:44.241 The truth table of XOR has not changed; A and C remain 00:02:44.262 --> 00:02:45.370 its true points. 00:02:46.499 --> 00:02:51.407 In a discrete topology every subset is both open and closed. 00:02:51.908 --> 00:02:56.661 XOR's interior and closure both equal its own truth region, so 00:02:56.672 --> 00:02:58.102 its boundary is empty. 00:02:58.594 --> 00:03:03.598 The Boolean event stayed fixed while its observational boundary changed. 00:03:04.525 --> 00:03:08.855 A point is on the boundary exactly when its smallest neighborhood meets 00:03:08.877 --> 00:03:11.199 both the event and its complement. 00:03:11.640 --> 00:03:16.035 The two witnesses lie in that same neighborhood: one inside the event 00:03:16.035 --> 00:03:17.316 and one outside. 00:03:17.811 --> 00:03:20.129 Neither witness has to be the centre. 00:03:20.538 --> 00:03:24.785 That neighborhood cannot certify either alternative completely. 00:03:25.867 --> 00:03:30.710 Symmetric difference compares two truth sets and removes their overlap. 00:03:31.242 --> 00:03:35.832 Boundary compares an event with the topology of available neighborhoods. 00:03:36.472 --> 00:03:41.897 Even a nonempty XOR region has empty boundary when the topology is 00:03:41.897 --> 00:03:46.040 discrete, so the two operations cannot be identified. 00:03:47.045 --> 00:03:52.477 When we replace positive X and Y by both signs, which changes: 00:03:52.671 --> 00:03:57.371 the XOR truth region, its interior, its closure, or its boundary? 00:03:57.922 --> 00:03:59.930 State the old and new boundary. 00:04:20.201 --> 00:04:23.186 The XOR truth region stays A and C. 00:04:23.870 --> 00:04:28.395 Its interior becomes A and C, its closure becomes A and C, 00:04:28.612 --> 00:04:31.012 and its boundary becomes empty. 00:04:31.208 --> 00:04:33.998 The earlier boundary was A, C and D. 00:04:34.747 --> 00:04:38.354 The interface changed, not classical evaluation. 00:04:38.418 --> 00:04:43.746 A boundary statement must identify its event, carrier and topology. 00:04:43.946 --> 00:04:50.261 Without those labels, a viewer may mistake Boolean disagreement for observational uncertainty 00:04:50.315 --> 00:04:54.189 or infer a physical edge that has no mathematical role in this 00:04:54.200 --> 00:04:55.531 finite example.