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 (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 *.


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