Let `Z` be the set of all integers and `R` be the relation on `Z` defined as `R={(a , b); a ,\ b\ in Z ,` and `(a-b)` is divisible by `5.}` . Prove that `R` is an equivalence relation.

Explanation

The given relation is `R={(a, b): a, b in 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.



Reflexive As for any `x in Z`, we have `x-x=0`, which is divisible by 5 .



`Rightarrow(x-x)` is divisible by `5 Rightarrow(x, x) in R, forall x in Z`



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