Question

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.

Answer 1

We have provided R = {(a, b) : a, b ∈ Z, and(a – b) is divisible by 5} 

(i) As (a – a) = 0 is divisible by 5. 

∴ (a, a) ∈ R ∀ a ∈ R 

Hence, R is reflexive. 

(ii) Let (a, b) ∈ R 

⇒ (a – b) is divisible by 5. 

⇒ – (b – a) is divisible by 5. 

⇒ (b – a) is divisible by 5. 

∴ (b, a) ∈ R 

Hence, R is symmetric. 

(iii) Let (a, b) ∈ R and (b, c) ∈ Z 

Then, (a – b) is divisible by 5 and (b – c) is divisible by 5. 

(a – b) + (b – c) is divisible by 5. (a – c) is divisible by 5. 

∴ (a, c) ∈ R 

⇒ R is transitive. 

Hence, R is an equivalence relation.