caption: S5 (cont'd) However, symmetry plus transitivity will do the job.
exif: {"Image DocumentName": "S5 (cont\u2019d) However, symmetry plus transitivity will do the job.", "Image ImageDescription": "Proof by contradiction: Suppose hw1(\u25cap \uf0ae \u25a1\u25cap) = false. Then hw1(\u25cap) = true and hw1(\u25a1\u25cap) = false. Since hw1(\u25a1\u25cap) = false, there must be a world w2 such that w1Rw2 and hw2(\u25cap) = false. And since hw1(\u25cap) = true, there must be a world w3 such that w1Rw3 and hw3(p) = true. Now, since R is symmetrical, we must have w2Rw1, and by transitivity, we must then have w2Rw3. But since hw2(\u25cap) = false, we must have hw3(p) = false. Contradiction. Of course, S5 adds the axiom \u25cap \uf0ae \u25a1\u25cap to the axioms of T and S4, requiring R to be reflexive, symmetrical, and transitive.", "Image XResolution": "37", "Image YResolution": "37", "Image ResolutionUnit": "Pixels/Centimeter", "Image Software": "SlidePlayer", "Image YCbCrPositioning": "Centered"}