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 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.

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 .

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

Show that the binary operation ** defined on RR by a**b=ab+2 is commutative but not associative.

Let RR be the set of real numbers. Show that the operation ** defined on RR-{0] by a**b=|ab|,a,binRR-{0} is a binary operation on RR-{0}.

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

(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?