UPSC NDA Math Model Paper 7

Given: R is a relation on Z and is defined as R = {(a, b): a - b is divisible by 7 where a, b ∈ Z}
R is reflexive as a - a = 0 is divisible by 7 for all a ∈ Z.
Suppose if (a, b) ∈ R ⇒ 7 divides a - b i.e a - b = 7m where m ∈ Z ⇒ b - a = 7n where n = - m ⇒ 7 divides b - a too which implies that (b, a) ∈ R.
Hence R is symmetric.
Suppose if (a, b) ∈ R and (b, c) ∈ R then a - b and b - c are divisible by 7.
⇒ a - b = 7m and b - c = 7n where m, n ∈ Z
⇒ a - c = 7q where q = m + n
⇒ (a, c) ∈ R
Hence R is transitive.
Thus relation R is an equivalence relation.