(d) An equivalence relationWe can check the given properties as follows: Reflexive: Let (a, b) ∈ N x N. Then (a, b) ∈ N ⇒ a + b = b + a (Communtative law of Addition) ⇒ (a, b) R (b, a) ⇒ (a, b) R (a, b) ⇒ R is reflexive. Symmetric: Let (a, b), (c, d) ∈ N x N such that(a, b) R (c, d). Then (a, b) R (c, d) ⇒ a + d = b + c ⇒ b + c = a + d ⇒ c + b = a + d (By commutativity of addition on N) ⇒ (c, d) R (a, b) ∴ R is symmetric. Transitive : Let (a, b), (c, d), (e, f) ∈ N x N such that (a, b) R (c, d) and (c, d) R (e, f). Then, ⇒ (a + d) + (c + f) = (b + c) + (d + e)⇒ a + c = b + e ⇒ (a, b) R (e, f)∴ (a, b) R (c, d) and (c, d) R (e, f) ⇒ (a, b) R (e, f) on N x N so R is transitive.Hence R is an equivalence relation on N × N.