Relation and Function

Relation: If there are two sets A and B. Then a relation R is subset of the Cartesian Product AxB. A={1,2,3} and B={a,b} then AxB={(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}. And relation R may be any subset of AxB such as R={(1,a), (1,b), (2,b), (3,a)} or R={(1,a), (2,b), (3,a)}.

A relation R from a set A to itself is a subset of set of the Cartesian Product AxA. Let A={1,2,3} then AxA={(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}. Then R may be any subset of AxA such as R={(1,1)} or R={(1,1), (2,2), (3,3)} or R=∅.

On basis of above example we try to understand three basic types of relation- Reflexive, Symmetric and Transitive.

Reflexive Relation: Relation R={(1,1), (2,2), (3,3)} is reflexive because all (a,a) elements are in set R where a is an element of A.

Symmetric Relation: R={(1,2), (2,1), (3,3)} or R={(1,1)} is symmetric Relation because if (a,b)∈R then (b,a)∈R for a,b ∈ A.

Transitive Relation:  R={ (1,3), (1,2), (2,3)} is transitive relation because if (a,b) and (b,c) are elements in R then (a,c) must also be an element in R for a,b,c ∈ A.

Function: A function is a relation such that every element of A is related to a unique element of B.