(i) A is said to be closed on * if all the elements of (a, b) *(c, d) = (a+ c, b+ d) belongs to N×N for A = N×N.

Let a = 1, b = 3, c = 8, d = 2

(1, 3) * (8, 2) = (1+8, 3+2)

= (9, 5) ∈N×N

Hence A is closed for *.

(ii) For commutative,

(c, d) *(a, b) = (c+ a, d+ b)

As addition is commutative a+ c = c+ a and b+ d = d+ b, hence * is commutative binary operation.

(iii) For associative,

(a, b) *((c, d) *(e, f)) = (a, b) *(c+ e, d+ f)

= (a+ c+ e, b+ d+ f)

((a, b) *(c, d)) *(e, f) = (a+ c, b+ d) *(e, f)

= (a+ c+ e, b+ d+ f)

As (a, b) *((c, d) *(e, f)) = ((a, b) *(c, d)) *(e, f), hence * is an associative binary operation.

(iv) For identity element (e₁, e₂), (a, b) *(e₁, e₂) = (e₁, e₂) *(a, b) = (a, b) in a binary operation.

(a, b) *(e₁, e₂) = (a, b)

(a+e₁, b+e₂) = (a, b)

(e₁, e₂) = (0, 0)

As (0,0) ∉N×N, hence identity element does not exist for *.