On the set `R-{-1}` a binary operation `*` is defined by `a*b=a+b+a b` for all `a , b in R-1{-1}` . Prove that * is commutative as well as associative on `R-{-1}dot` Find the identity element and prove that every element of `R-{-1}` is invertible.

Explanation

We have
`a^{*} b=a+b+a b` for all `a, b in A,` where `A=R-{-1}`

Commutativity: For any `a, b in R-{-1}`

To prove:` a^{*} b=b^{*} a`

Now, `a * b=a+b+a b`

`b^{*} a=b+a+a b`

From (1) and (2), we get


`a^{x} b=b^{*} a`

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