Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation.

Dear Student

Given
∀a, b∈Z, aRb if and only if a–b is divisible by n.

aRa⇒(a -a) is divisible by n, which is true for any integer a as ‘0’ is divisible by n.Thus, R is reflective.

Now, aRb
(a - b) is divisible by n.
⇒- (b - a) is divisible by n.
⇒(b–a) is divisible by n
⇒bRa
Thus, R is symmetric.

Let aRb and bRc
Then, (a - b) is divisible by n and (b - c) is divisible by n.
So, (a - b) + (b - c) is divisible by n.
⇒(a - c) is divisible by n.
⇒aRc
Thus, R is transitive.

Thus, R is an equivalence relation.

Thanks