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 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 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.

Let @ be an operation defined on RR . The set of real numbers, by a@b=min(a,b) for all a,binRR . Show that @ is a binary operation on RR .

Show that **: R xx R to R given by (a,b) to a + 4b ^(2) is a binary operation.

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

The remainder when (sum_(k=1)^(5) ""^(20)C_(2k-1))^(6) is divided by 11, is :

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