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On R-[1] , a binary operation * is defin...

On `R-[1]` , a binary operation * is defined by `a*b=a+b-a b` . Prove that * is commutative and associative. Find the identity element for * on `R-[1]dot` Also, prove that every element of `r-[1]` is invertible.

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We have
` a^{*} b=a+b+a b for a , b in A, where A=R-{-1}`
Commutativity. For any `a, b in R-{-1}`
To prove:` a^{*} b=b^{*} a`
Now, `a^{2} b=a+b+a b`
`b^{*} a=b+a+a b`
From (1) and (2), we get
`a^{*} b=b^{*} a`
...
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