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.