Question

Show that the relation R in the set of integers Z, defined as xRy ⇒ x – y, is divisible by 5, where x, y ∈ Z, is an equivalence relation.

Answer 1

(i )Reflexive: ∀ x ∈ Z, x – x = 0 and (x – x)/5 = 0

i.e.r x – x, is divisible by 5.

⇒ xRx (reflexive).

So, R is reflexive.

(ii) Symmetric: Let ∀ x, y ∈ Z, xRy is true, then

xRy ⇒ x – y, is divisible by 5

⇒ y – x, is divisible by 5

[∵(y – x) = – (x – y)]

⇒ yRx (symmetric)

So, R is symmetric.

(iii) Transitive: Let ∀x,y,z ∈ Z, xRy and yRz is true so xRy and yRz ⇒ (x – y) and

(y – z) both are divisible by 5.

⇒ (x – y) + (y – z) also divisible by 5.

⇒ x – z, is divisible by 5.

⇒ xRz (transitive)

So, R is transitive.

Thus, given relation is an equivalence relation.