Let * be an associative binary operation on a set S with the identity element e in S. Then. the inverse of an invertible element is unique.

Define an associative binary operation on a set.

A binary operation on a set has always the identity element.

Let * be an associative binary operation on a set S and a be an invertible element of S. Then; inverse of a^(^^)-1 is a.

Let * be a binary operation on Z defined by a*b=a+b-4 for all a,b in Z. Find the identity element in Z .(ii) Find the invertible elements in Z .

Definition and Theorem: Let * be a binary operation on a set S.If S has an identity element for * then it is unique.

Determine the total number of binary operations on the set S={1,2} having 1 as the identity element.

Write the total number of binary operations on a set consisting of two elements.

For the binary operation xx10 on set S={1,3,7,9}, find the inverse of 3.

Consider the set S={1,-1} of square roots of unity and multiplication (x) as a binary operation on S .Construct the composition table for multiplication (xx) on S Also,find the identity element for multiplication on S and the inverses of various elements.