Answer 1

R = {(a, b)/|a − b| is even, a, b ∈ A}, where

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

|a − a| = 0 is even

∴ aRa ∀ a ∈ A

∴ R is reflexive

Let aRb

∴ |a − b| is even

∴ |a − b| = |b − a|

∴ |b − a| is even

∵ bRa

∴ aRb ⇒ bRa ∀a, b ∈ A

∴ R is symmetric

Let aRb and bRc

∴ |a − b| and |b − c| are even

If b is even, then a and c both are even

∴ |a − c| is even

If b is odd, then a and c both are odd

∴ |a − c| is even

∴ aRb, bRc ⇒ aRc ∀a, b, c ∈ A

∴ R is transitive

∵ R is reflexive, symmetric, and transitive

∴ R is an equivalence relation