We have relation defined on the set N x N defined by

(a, b) R(c, d) if a² + d² = b² + c² ∀ a, b, c, d ∈ N.

Reflexivity : Since, a² + b² = a² + b² , where a, b ∈ N. 

⇒ a² + b² = b² + a² ( \(\because\) Addition on natural numbers is commutative.)

⇒ (a, b) R (a, b) ∀ a, b ∈  N.

Hence, relation R is reflexive relation.

Symmetricity : Let a, b, c, d ∈ N and (a, b) R (c, d)

⇒ a² + d² = b² + c²

⇒ b² + c² = a² + d²

⇒ c² + b² = d² + d² ( \(\because\) Addition on natural numbers is commutative. )

⇒ (c, d) R (a, b) ∀  a, b, c, d ∈ N.

Hence, relation R is symmetric relation.

Transitivity : Let a, b, c, d, e, f ∈ N, (a, b) R (c, d) and (c, d) R(e, f).

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

⇒ a² + d² = b² + c² and c² + f² = d² + e²

⇒ a² + d² + c² + f² = b² + c² + d² + e² (Adding two equations )

⇒ a² + f² = b² + e² (Cancellation law of addition on N)

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

Hence, relation R is transitive relation.

Since relation R is reflexive, symmetric and transitive relation.

Therefore, relation R is an equivalence relation.