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 ** 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 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 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 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.

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.

An operation '*' is defined on a set A={1,2,3,4} as follows: a*b=ab(mod5), all a,b in A ,Prepare the composition table for '*' on A and from the table show that, '*' is a binary opration and '*' is commutative on A.

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

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 .

Prove that the operation @ on QQ , the set of rational numbers, defined by a@b=ab+1 is binary operational on QQ .

(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?