section 4.4 properties of relations. order relations draw an arrow diagram for the relation r...
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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:
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