Section 4.4

Properties of Relations

Order Relations

Draw an arrow diagram for the relation R defined on the set {1,2,3,4} such that }:),{( yxyxR

1 2

3 4

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

Rab ),(Rba ),(Rba , ba

Rba ),( Rcb ),(

Rca ),(

Raa ),( Aa

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?

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?

Define a relation R on Z as follows: is even}

Is R a partial order?

baZxZbaR :),{(

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.

Raa ),( Aa

Practice:

Practice:

Practice: