CISC 203 Chapter Notes - Chapter 54: Partially Ordered Set, Transitive Relation, Hasse Diagram

• School: Queen's University
• Department: Computing
• Course: CISC 203
• Professor: Robin W Dawes

Document Summary

Chapter 54: fundamentals of partially ordered sets (textbook notes) A pair, =(cid:4666),(cid:4667) where x is a non-empty set and r is a relation on x that satisfies: R is antisymmetric ( (cid:1876),(cid:1877), , (cid:1876)(cid:1877) (cid:1877)(cid:1876) then (cid:1876)=(cid:1877)) X is (cid:272)alled the (cid:862)g(cid:396)ou(cid:374)d set(cid:863) of p. the ele(cid:373)e(cid:374)ts of x a(cid:396)e the ele(cid:373)e(cid:374)ts of the pa(cid:396)tiall(cid:455)-ordered set. R is (cid:272)alled the pa(cid:396)tial o(cid:396)de(cid:396) (cid:396)elatio(cid:374). (cid:862)poset(cid:863) is a(cid:374) a(cid:271)(cid:271)(cid:396)e(cid:448)iatio(cid:374) of (cid:862)pa(cid:396)tiall(cid:455) o(cid:396)de(cid:396)ed set(cid:863) Fo(cid:396) (cid:373)ost posets, it"s eas(cid:455) to (cid:272)he(cid:272)k for the reflexive and antisymmetric properties (the only time we have both (x, y) and (y, x) in r is when x = y). How to draw a diagram to depict a poset. Each element of the ground set (x) is represented by a dot. If (cid:454) r (cid:455) i(cid:374) the poset, the(cid:374) d(cid:396)a(cid:449) (cid:454)"s dot (cid:271)elo(cid:449) (cid:455)"s dot a(cid:374)d d(cid:396)a(cid:449) a seg(cid:373)e(cid:374)t f(cid:396)o(cid:373) (cid:454) to (cid:455) No need to draw line between a dot and itself.