A = {1, 2, 3, 4, 5} R = {(a, b): a - b is even} For R to be an equivalence relation it must be (i) Reflexive, |a - a| = 0 ∴ (a,a) ∊ R for ∀ a ∊ A So R is reflexive. (ii) Symmetric if (a,b) ∊ R ⇒ |a - b| is even ⇒ |b- a| is also even So R is symmetric. (iii) Transitive If (a, b) ∊ R (b, c) ∊ R then (a, c) ∊ R (a, b) ∊ R ⇒ |a - b| is even (b, c) ∊ R ⇒ |b - c| is even Sum of two even numbers is even So, |a - b| + |b - c| |a - b + b - c| = |a - c| is even since, |a - b| and |b - c| are even So (a ,c) ∊ R Hence, R is transitive. Therefore, R is an equivalence relation.