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}.

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

State whether the following statements are true or false, Justify. If ** is a commutative binary opertion on N, then a ** (b **c) = (c**b) **a

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

State whether the following statements are true or false with reasons. (i) For any binary operation ** on NN , a**a=aAAainNN . (ii) If ** , a binary operation ** on NN is commutative then, a**(b**c)=(c**b)**a .

Let A={1,omega,omega^(2)} be the set of cube roots of unity. Prepare the composition table for multiplication (xx) on A. Show that multiplication on A is a binary operation and it is commutative on A. Find the identity element for multiplication and show that every element of A is invertible.

Let S={1,omega,omega^(2)} be the set of cube roots of unity. Prepare the composition table for multiplication (xx) on S, show that multiplication on S is a binary operation and it is commutative on S. Also, show that every element on S is invertible.

Determine which of the following binary operations are associative and which are commutative: (i) ** on RR defined by a**b=1 for all a,binRR . (ii) ** on RR defined by a**b=(a+b)/(2) for all a,binRR .

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.

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