Question

If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and, a - b is divisible by 5}. Prove that R is an equivalence relation.

Answer 1

The given relation is R = {(a, b):a,b ∈ Z and a-b is divisible by 5}.

To prove R is an equivalence relation, we have to prove R is reflexive, symmetric and transitive.

Therefore, R is transitive. 

Thus, R is reflexive, symmetric and transitive. 

Hence, it is an equivalence relation.