## Saturday, December 20, 2008

### Types of relations

Void or empty relation
When A is a set, ф is a subset of A×A and so it is a relation on A. This relation is called the void or empty relation on A.

Universal relation
When A is a set, A×A is a subset of A×A and so it is a relation on A. This relation is called the universal relation.

Identity relation
When A is a set, the relation IA = {(a,a):a ЄA} on A is called the identity relation on A.

Reflexive relation
A relation on a set A is said to be reflexive if every element of A is related to itself.

Think. What is the difference between identity relation and reflexive relation?

Symmetric relation
A relation R on a set A is said to be a symmetric relation iff

(a,b) Є R implies (b,a) ЄR for all a,b Є A.

i.e. aRb implies bRa for a,b Є.

Transitive relation
A relation R on a set A is said to be a transitive relation iff
(a,b) ЄR and (b,c) ЄR implies (a,c) ЄR for a,b,c ЄA.

Antisymmetric relation
A relation R on set A is said t be an antisymmetric relation iff
(a,b) Є R and(b,a) Є R implies a =b for all a,b ЄA.

Equivalence relation
A relation R on set A is said to be an equivalence relation on A iff
i. it is reflexive.
ii. it is symmetric
iii. it is transitive.