Answer :

**(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 (e_{1}, e_{2}), (a, b) *(e_{1}, e_{2}) = (e_{1}, e_{2}) *(a, b) = (a, b) in a binary operation.

(a, b) *(e_{1}, e_{2}) = (a, b)

(a+e_{1}, b+e_{2}) = (a, b)

(e_{1}, e_{2}) = (0, 0)

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

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