The relation R on the set A = {1, 2, 3, 4, 5}, given by R{(a, b) : |a - b| is even}, is :

Equivalence

A = {1, 2, 3, 4, 5}

R : {(a, b) : |a - b| is even}

so for  a, a ∈ A

|a - a| = 0 (even) ⇒ reflective

since |a - b| = |b - a|  for  (a, b) ∈ R

(b, a) ∈ R ⇒ symmetric

so |a - b| → even      (a, b) ∈ R

|b - c| → even      (b, c) ∈ R

true evenly when a, b both even or both are odd

if b is odd c is also odd

b is even c is also even

⇒ (a, c) both are even/odd

so  |a - c| → even

|a - c| ∈ (R) 

so its transitive

hence its an equivalence relation