if * is a binary operation on R defined by a*b= a+b+ab. prove that * is commutative and associative. find the idebtify element. also show that every element of R is invertible wxcept -1.

Given, * is a binary operation on R defined by a*b=a+b-ab

Let . Then

Thus, So, * is a binary operation on R.

Now,

Commutativity : , we have,

So, * is commutative on R

Associativity : , we have,

So, * is associative on R

Identity Element : Let e be the identity element in R, then

So, 0 is the identity element in R.

Inverse of an Element : Let a be an arbitrary element of R and b be the inverse of a. Then,

This inverse exist only if . So, every element of R is invertible except -1.

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