Correct Answer is (D) an equivalence relation

Given set Z = {1, 2, 3 ,4 …..}

And R = {(a, b) : a,b ∈ Z and (a-b) is divisible by 3}

Formula

For a relation R in set A

Reflexive

The relation is reflexive if (a , a) ∈ R for every a ∈ A

Symmetric

The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R

Transitive

Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R

Equivalence

If the relation is reflexive , symmetric and transitive , it is an equivalence relation.

Check for reflexive

Consider , (a,a)

(a - a) = 0 which is divisible by 3

(a,a) ∈ R where a ∈ Z

Therefore , R is reflexive ……. (1)

Check for symmetric

Consider , (a,b) ∈ R

∴ (a - b) which is divisible by 3

- (a - b) which is divisible by 3

(since if 6 is divisible by 3 then -6 will also be divisible by 3)

∴ (b - a) which is divisible by 3 ⇒ (b,a) ∈ R

For any (a,b) ∈ R ; (b,a) ∈ R

Therefore , R is symmetric ……. (2)

Check for transitive

Consider , (a,b) ∈ R and (b,c) ∈ R

∴ (a - b) which is divisible by 3

and (b - c) which is divisible by 3

[ (a-b)+(b-c) ] is divisible by 3 ] (if 6 is divisible by 3 and 9 is divisible by 3 then 6+9 will also be divisible by 3)

∴ (a - c) which is divisible by 3 ⇒ (a,c) ∈ R

Therefore (a,b) ∈ R and (b,c) ∈ R then (a,c) ∈ R

Therefore , R is transitive ……. (3)

Now , according to the equations (1) , (2) , (3)

Correct option will be (D)