Given,
R = {(a, b) : a, b ∈ Z and (a – b) is divisible by 5}
Reflexivity : ∀ a ∈ Z
a – a = 0 is divisible by 5
⇒ (a, a) ∈ R ∀ a ∈ Z
Hence, R is reflexive.
Symmetry : Let (a, b) ∈ R
⇒ a – b is divisible by 5
⇒ – (b – a) is divisible by 5
⇒ (b – a) is divisible by 5
⇒ (b, a) ∈ R
Hence, R is symmetric.
Transitivity : Let (a, b), (b, c) ∈ R
⇒ (a – b) and (b – c) are divisible by 5
⇒ (a – b + b – c) is divisible by 5
⇒ a – c is divisible by 5
⇒ (a, c) ∈ R
Thus, R is an equivalence relation.