Let `S=(0,1,2,3,4,)` and * be an operation on S defined by `a*b=r` , where`r` is the least non-negative remainder when `a+b` is divided by 5. Prove that * is a binary operation on S.

Let S={0,1,2,3,4}, if a,binS , then an operation ** on S is defined by, a**b=r where r is the non-negative remaider when (a+b) is divided by 5. Prove that ** is a binary operation on S.

Let ** be an opeartion defined on S={1,2,3,4} by a**b=m where m is the least non-negative remainder when the product ab is divided by 5. Prove that ** is a binary operation on S.

Let A={0,1,2,3,4,5}. If a_(1)binA, then an operation @ on A is defined by a@b=k where k is the least non-negative remainder when the sum (a+b) is divided by 6. Show that @ is a binary operation on A.

Let ** be an operation defined on A,={2,4,6,8} by a**b=k where k is the least non-negative remainder when the product ab is divided by 10 and a,binA. show that ** is a binary operation on A.

Let ** be the binary defined on the set S={1,2,3,4,5,6} by a**b=r where r is the least non-negative remainder when ab is divided by 7. Prepare the composition table ** on S. Observing the composition table show that 1 is the identity element for ** and every element of S is invertible.

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 ** be an operation defined on NN , the set of natural numbers, by a**b=L.C.M.(a,b) for all a,binNN . Prove that ** is a binary operation on NN .

The operation @ is defined by a@b=b^(a) on the set Z={0,1,2,3,…} . Prove that @ is not a binary opeartion on Z.

Prove that the operation ** on ZZ defined by a**b=a|b| for all a,binZZ is a binary operation

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.