Let A= `{a+sqrt5b:a,binZ}`. Show that usual multiplication of numbers is binary operation on 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.

An operation ** is defined on the set A={1,2,3,4} as follows: a**b,ab(mod5) for all a,binA .Prepare the compositon table for ** on A and from the table show that - (i) multiplication mod(5) is a binary operation, (ii) ** is commutative on A, (iii) is the identity element for multiplication mod(5) on A, and. (iv) every element of A is invertible.

Let A={1,-1} be the set of square roots of unity. Considering multiplication (xx) as a binary operation on A, construct the composition table for multiplication on A. Determine the identity element for multiplication in A and the inverses of the elements.

If M_(2) be the set of all 2xx2 matrices of the form {:((a,a),(a,a)):}, where ainR-{0} , then the identity element with respect to the multiplication of matrices as binary operation, is--

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.

Prove that the operation 'addition' on the set of irrational numbers is not a binary operation.

If the binary oprations ** on RR is defined by a**b=a+b+ab for all a,binRR where on R.H.S. we have usual addition, subtraction and multiplication of real numbers. The relation ** is---

Number of binary opertions on the set {a,b} are

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 .