Let Z be the set of all integers and R be 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.

Given R = {(a, b): 5 divides (a – b)}

For equivalence relation we have to check three parameters:

(i) Reflexive:

If (a-b) is divisible by 5 then,

⇒ (a-a) =0 is also divisible by 5

⇒ (a, a) ∈ R

Hence R is Reflexive ∀ (a,b) ∈ Z

(ii)Symmetric:

If (a-b) is divisible by 5 then,

⇒ (b-a) =-(a-b) is also divisible by 5

⇒ (a,b) ∈ R and (b,a) ∈ R

Hence R is Symmetric ∀ (a,b) ∈ Z

(iii)Transitive:

If (a-b) and (b-c) are divisible by 5 then,

⇒ a-c=(a-b)+(b-c) is also divisible by 5

⇒ (a,b) ∈ R , (b,c) ∈ R and (a,c) ∈ R

Hence R is Transitive ∀ (a,b) ∈ Z

⇒ As Relation R is satisfying all the three parameters, hence R is an equivalence relation.