1. Section 4.4 Properties of Relations

2. Order Relations Draw an arrow diagram for the relation R defined on the set {1,2,3,4} such that 1 2 3 4

3. Definition: Let R be a binary relation on A. R is reflexive if for all R is antisymmetric if for all, if and then R is transitive if whenever and it must also be the case that

4. Definition A relation R on a set A is called a partial order on A if R is antisymmetric, transitive, and reflexive. Exercise: Is the previous relation a partial order?

5. Let A:= P ({1,2,3}) and define a relation R on A such that s R t if n(s  t) = . Is R a partial order?

6. Define a relation R on Z as follows: is even} Is R a partial order?

7. Definition: R is irreflexive if for all A strict partial ordering on a set A is a relation R on A that is transitive, irreflexive, and antisymmetric.

8. Practice: