Answer :

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


i) Commutative


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


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


For all a,b,c,d ϵ R


*is commulative on A


II) Associative:


(a,b),(c,d),(e,f) ϵ A


{(a,b)*(c,d)}*(e,f)


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


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


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


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


= (a*b){(c,d)*(e,f)}


Is associative on A


Let (x,y) be the identity element in A,


Then,


(a,b)*(x,y) = (a,b) for all (a,b)ϵ A


(a + x,b + y) = (a,b) for all (a,b)ϵ A


(a + x = a,b + y = b) for all (a,b)ϵ A


X = 0 ,y = 0


(0,0)ϵ A


(0,0) is the identity element in A


Let (a,b) be an invertible element of A.


(a,b)*(c,d) = (0,0) = (c,d)*(a,b)


(a + c,b + d) = (0,0) = (c + a,b + d)


a + c = 0 b + d = 0


a = - c b = - d


c = - a d = - b


(a,b) is invertible


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