Given : Set X = {(a, b) : a, b ∈ I} 

where I is the set of integers. 

A relation R in X is defined as: 

(a, b) R(c, d) ⇔ b – d = a – c ∀ (a, b) (c, d) ∈ X 

To prove that R is equivalence relation, we have to prove that R is reflexive, symmetric and transitive

(i) Reflexivity: 

Let (a, b) ∈ X 

(a, b) ∈ X ⇒ (a, b) ∈ I 

⇒ b – b = a – a = 0 

⇒ (a, b) R(a, b) ∀ (a, b) ∈ X 

R is a reflexive relation. 

(ii) Symmetricity: 

Let (a, b), (c, d) ∈ X is in this way 

(a, b) R(c, d) 

(a, b) R(c, d) 

⇒ b – d = a – c 

⇒ -(d – b) = -(c – a) 

⇒ d – b = c – a 

⇒ (c.d) R(a.b) 

(a, b) R(c, d) ⇒ (cd) R(ab) ∀ (a, b), (c, d) ∈ X 

R is a symmetric relations. 

(iii) Transitivity: 

Let (a, b), (c, d), (e, f) ∈ X is in this way 

(a, b) R(c, d) and (c, d) R(e, f) (a, b) R(c, d) 

⇒ b – d = a – c …(1) 

(c, d) R(e, f) ⇒ d – f = c – e …(2) 

Adding equation (i) and (2), we have 

b – d + d – f = a – c + c – e 

⇒ b – f = a – e 

⇒ (a, b) R(e, f) 

So, (a, b) R(c, d) and (c, d) R(e, f) 

⇒ (a, b) R(e, f) ∀ (a, b), (c, d), (e, f) ∈ X 

R is a transitive relation. 

Hence, according to (i), (ii) and (iii), the given relation is equivalence relation. 

Hence Proved.