Let R be a relation on set A . If R is reflexive, symmetric, and transitive then it is said to be equivalence relation. Consequently, two elements a and b related by an equivalence relation are said to be equivalent .
Example - Show that relation R ={(a,b)|a=b(mod m)} is an equivalence relation.
Solution -
- Reflexive - for element a-a=0 is divisible by m.
a=a(mod m) . So congruence modulo m is reflexive.
- Symmetric - For any two elements a and b , if (a,b) €R i.e. Congruence Modulo m is Symmetric.
- Transitive - For any three elements a,b and c if (a,b), (b,c) € R then
(a-b) mod m =0
(b-c) mod m =0
Adding both above equations (a-c)mod m =0
So R is transitive
Therefore R is an equivalence relation