WEBVTT 00:00:00.250 --> 00:00:03.848 Our original cards use X for dot and Y for stripe. 00:00:04.257 --> 00:00:08.074 A has only a dot, B both, C only a stripe, and 00:00:08.085 --> 00:00:09.021 D neither. 00:00:09.293 --> 00:00:12.355 The two-bit annotations follow X,Y order. 00:00:12.875 --> 00:00:18.203 Selecting just one question keeps only that answer in the selected description; 00:00:18.409 --> 00:00:21.323 it leaves the complete card unchanged. 00:00:22.274 --> 00:00:25.649 Observe only X on our four valuations. 00:00:26.279 --> 00:00:31.151 A and B share the selected description true; C and D share 00:00:31.151 --> 00:00:33.304 the selected description false. 00:00:33.769 --> 00:00:38.610 The complete states still differ, but the chosen description groups them into 00:00:38.620 --> 00:00:40.400 two equivalence classes. 00:00:41.388 --> 00:00:46.676 An equivalence class collects points related by this equality of descriptions. 00:00:47.230 --> 00:00:50.372 Classes do not overlap and cover the carrier. 00:00:50.847 --> 00:00:55.807 Replacing each class by one label is a later operation; today we 00:00:55.817 --> 00:00:57.868 keep track of the grouping itself. 00:00:59.021 --> 00:01:01.653 Now select both X and Y. 00:01:02.171 --> 00:01:07.596 Instead of grouping equal descriptions, collect the selected formulas true at each 00:01:07.607 --> 00:01:08.178 point. 00:01:08.565 --> 00:01:11.364 This collection is its positive support. 00:01:11.778 --> 00:01:17.020 D has none, A has only X, C has only Y, and 00:01:17.030 --> 00:01:18.095 B has both. 00:01:18.618 --> 00:01:24.086 Here each formula occurs once; later we will use position labels when 00:01:24.086 --> 00:01:27.016 a selected list can repeat a formula. 00:01:27.975 --> 00:01:33.433 The description containing X is included in the description containing X and 00:01:33.487 --> 00:01:33.911 Y. 00:01:34.434 --> 00:01:38.914 Every positive item in the first remains present in the second. 00:01:39.464 --> 00:01:46.229 Inclusion expresses a one-way comparison, not equality between the two descriptions. 00:01:47.363 --> 00:01:52.242 Put the empty support at the bottom, the two singleton supports in 00:01:52.242 --> 00:01:55.702 the middle, and the two-item support at the top. 00:01:56.240 --> 00:02:00.343 Draw upward links only when one item is added. 00:02:00.887 --> 00:02:07.067 The diagram records inclusion, not physical or numerical distance. 00:02:07.972 --> 00:02:10.799 The two middle supports are incomparable. 00:02:11.295 --> 00:02:13.563 Each has something the other lacks. 00:02:14.172 --> 00:02:19.454 Different supports need not be comparable, so this diagram is not a 00:02:19.507 --> 00:02:25.447 single line ranking points from universally worse to universally better. 00:02:26.432 --> 00:02:28.982 Every support includes itself. 00:02:29.626 --> 00:02:34.180 A relation with this self-comparison property is called reflexive. 00:02:34.599 --> 00:02:39.048 We can record comparisons in a table: the row is the first 00:02:39.069 --> 00:02:43.023 object and the column is the proposed second object. 00:02:43.971 --> 00:02:48.345 If every item in the first support occurs in the second, and 00:02:48.399 --> 00:02:52.432 every item in the second occurs in the third, then every item 00:02:52.432 --> 00:02:54.478 in the first occurs in the third. 00:02:54.897 --> 00:02:58.143 This chaining property is called transitive. 00:02:58.829 --> 00:03:03.690 An upper set keeps every description above any description it contains. 00:03:04.294 --> 00:03:09.102 If it contains the support with X alone, it must also contain 00:03:09.102 --> 00:03:11.530 the support with both X and Y. 00:03:12.056 --> 00:03:15.857 We check this by following the inclusion links upward. 00:03:16.802 --> 00:03:22.618 An antichain is a collection of distinct supports with no containment comparison 00:03:22.640 --> 00:03:24.152 between different members. 00:03:24.738 --> 00:03:27.982 Our two middle supports form an antichain. 00:03:28.587 --> 00:03:32.927 This is different from an upper set: neither term means merely a 00:03:32.991 --> 00:03:35.424 random subset of the diagram. 00:03:36.404 --> 00:03:41.101 Under the one-test menu X, compare A with B, and then 00:03:41.133 --> 00:03:42.002 D with A. 00:03:42.611 --> 00:03:45.335 Which pair has equal descriptions? 00:03:45.974 --> 00:03:50.086 For which pair does inclusion work in just one direction? 00:04:06.747 --> 00:04:11.817 A and B have equal descriptions, so each support includes the other. 00:04:12.375 --> 00:04:17.225 D has the empty support and A has the support containing X. 00:04:17.904 --> 00:04:22.137 The empty support is included in A's support, but the reverse 00:04:22.137 --> 00:04:23.396 inclusion fails. 00:04:24.530 --> 00:04:28.350 Equality of descriptions groups points into classes. 00:04:28.994 --> 00:04:33.241 Inclusion orders the descriptions, possibly asymmetrically. 00:04:33.812 --> 00:04:39.140 Later we will prove that this inclusion direction is exactly the specialization 00:04:39.150 --> 00:04:42.354 convention used by the observation topology.