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.
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.