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.

Define an associative binary operation on a set.

Define a binary operation on a set.

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 a commutative binary operation on a set.

Associativity of binary operations

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

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

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

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

Define a binary operation * on the set A={0,1,2,3,4,5} as a*b=a+b( mod 6). Show that zero is the identity for this operation and each element a of the set is invertible with 6-a being the inverse of a.