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

R is an equivalence relation

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

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

It is clear that for any element a ∈ A, we have |a - a|= 0 (which is even).

Therefore, R is reflexive.

Let (a, b) ∈ R.

⇒ |a - b| is even,

⇒ |-(a - b)| = |b - a| is also even

⇒ (b, a) ∈ R

Therefore, R is symmetric.

Now, let (a, b) ∈ R and (b, c) ∈ R.

⇒ |a - b| is even and |b - c| is even

⇒ (a - b) is even and (b - c) is even  (assuming that a > b > c)

⇒ (a - c) = (a - b) + (b - c) is even  [Sum of two even integers is even]

⇒ |a - c| is even

⇒ (a, c) ∈ R

Therefore, R is transitive.

Hence, R is an equivalence relation.