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

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

Now, R = {(a, b): |a – b| is even}

Reflexive:

Let a ∈ A, then |a – a| = 0, which is even, Hence (a, a) ∈ R

Therefore, R is reflexive

Symmetric:

Let (a, b) ∈ R

⇒ |a – b| is even

⇒ |b – a| is even (∵ |a – b| = |b – a|)

⇒ (b, a) ∈ R

⇒ R is symmetric

Transitive:

Let (a, b) ∈ R and (b, c) ∈ R

⇒ |a – b| and |b – c| are even

Now,

|a – c| = |(a – b) + (b – c)|

As, sum of two even numbers is also even

⇒ |a – c| is even

⇒ R is transitive

As, R is reflexive, symmetric and transitive

⇒ R is an equivalence relation