Equivalence relation:
A relation which is reflexive, symmetric and transitive is called an equivalence relation.

Illustrative examples:
Ex. 1: Let  be a relation on , defined by

Show that  is an equivalence relation.
Solution: Given  and 
i) Let  then 

So,  is reflexive.

ii) 
i.e.  is an integer
 is an integer
 is an integer

Thus 
 is symmetric.
iii) (a, b)  and 
 is an integer and  is an integer
 is an integer
 is an integer

Thus  and 
 is transitive.

Thus,  is reflexive, symmetric and transitive.
 is an equivalence relation.