Question

Prove that the relation R on Z, defined by R = {(x, y) : (x – y) is divisible by 5} is an equivalence relation.

Answer 1

The given relation is R = {(x, y) : x, y ∈ Z and x – y 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 ∈ Z, we have x – x = 0, which is divisible by 5.

⇒ (x – x) is divisible by 5

⇒ (x, x) ∈ R, ∀ x ∈ Z

Therefore, R is reflexive.

Symmetric Let (x, y) ∈ R, where x, y ∈ Z

⇒ (x – y) is divisible by 5   ...[By definition of R]

⇒ x – y = 5A for some A ∈ Z

⇒ y – x = 5(–A)

⇒ (y – x) is also divisible by 5

⇒ (y, x) ∈ R

Therefore, R is symmetric.

Transitive Let (x, y) ∈ R, where x, y ∈ Z

⇒ (x – y) is divisible by 5

⇒ x – y = 5A for some A ∈ Z

Again, let (y, z) ∈ R, where y, z ∈ Z

⇒ (y – z) is divisible by 5

⇒ y – z = 5B for some B ∈ Z

Now, (x – y) + (y – z) = 5A + 5B

⇒ x – z = 5(A + B)

⇒ (x – z) is divisible by 5 for some (A + B) ∈ Z

⇒ (x, z) ∈ R

Therefore, R is transitive.

Thus, R is reflexive, symmetric and transitive.

Hence, it is an equivalence relation.