Let `A = N xx N and **` be the binary opertion on A defined by
` (a,b) ** (c,d) = (a + c, b+d)`
Show that `**` is commutative and associative.

Let S=NNxxNNand** is a binary operation on S defined by (a,b)**(c,d)=(a+c,b+d) for all a,b,c,d in NN . Prove that ** is a commutative and associative binary operation on S.

Let A=N x N and *be a binary operation on A defined by (a,b)*(c,d) =(a+c,b+d).Show(A*) has on identity element.

Let S = N xx N and ast is a binary operation on S defined by (a,b)^**(c,d) = (a+c, b+d) for all a,b,c,d in N .Prove that ** is an associate binary operation on S.

Let A=NNxxNNand@ be a binary operation on A defined by (a,b)@(c,d)=(ac,bd) for all a,b,c,dinNN . Discuss the commutativity and associativity of @ on A.

Let * be a binary operation on Zdefined by a*b=a+b- 15 for a,binZ . Show that *is commutative

(I) Let ** be a binary operation defined by a**b=2a+b-3. Find 3**4. (ii) let ** be a binary operation on RR-{-1} , defined by a**b=(1)/(b+1) for all a,binRR-{-1} Show that ** is neither commutative nor associative. (iii) Let ** be a binary operation on the set QQ of all raional numbers, defined as a**b=(2a-b^(2)) for all a,binQQ . Find 3**5 and 5**3 . Is 3**5=5**3?

Show that the operation ** defined on RR-{0} by a**b=|ab| is a binary operation. Show also that ** is commutative and associative.

Let A=Ncup{0}xxNNcup{0}, a binary operation ** is defined on A by. (a,b)**(c,d)=(a+c,b+d) for all (a,b),(c,d)inA. Prove that ** is commutative as well as associative on A. Show also that (0,0) is the identity element In A.

Prove that the binary operation ** on RR defined by a**b=a+b+ab for all a,binRR is commutative and associative.

Prove that the binary operation @ defined on QQ by a@b=a-b+ab for all a,b in QQ is neither commutative nor associative.