Define an associative binary operation on a non-empty set S.

Define a commutative binary operation no a non-empty set A.

Define a binary opeartion ** on a non-empty set A.

Let ** and @ be two binary operations on a non-empty setA. Then write the condition for which the binary operation ** is distibutive over binary operation @

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 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.

Discuss the commutativity and associativity of binary operation ** defined on ZZ by the rule a**b=|a|b for all a,binZZ .

Complete the following multiplication table so as to define a commutative binary operation ** on S={a,b,c,d}

Complete the following nultiplication table so as to define a commutative binary operation ** on A={1,2,3,4}.

(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?

Let S be a non-empty set and P(S) be the power set of the Set S. Statement -I: Phi is the identity element for union as a binary operation on P(S) Statement -II: S is the identity element for intersection on P(S).