Question

A relation R is defined on Z, the set of integers, as,

R = {(x, y): |x − y| is divisible by a prime number ‘p’, x, y ∈ Z}.

Check whether R is an equivalence relation or not.

Answer 1

Given:

R = {(x, y): |x − y| is divisible by a prime p}, this means x − y is divisible by p.

⇒ Reflexive:

|x − x| = 0, and 0 is divisible by p. So, R is reflexive.

⇒ Symmetric:

If |x − y| is divisible by p, then |y − x| is also divisible by p. So, R is symmetric.

⇒ Transitive:

If p | (x − y) and p | (y − z), then

p | [(x − y) + (y − z)] |

p | (x − z)

So, R is transitive.

Hence, R is an equivalence relation.