made easy

Question

A binary operator ⊕ on a set of R – {–1} is defined as x ⊕ y = x + y + xy. Which of the following statement is true about

(S, ⊕)?

A)(S, ⊕) is group but not abelian group

B)(S, ⊕) is monoid but not group

C)(S, ⊕) is semi-group but not monoid

D)(S, ⊕) is abelian group

Comments

is it associative ?

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it is associative

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$D$ ??

Answer 1

Ans is $D.$

Here $\oplus$ operation is associative. Therefore it is semi-group.

Further there exist an identity e for every element. $e= 0$, since $a \oplus 0 = a$, Therefore it is monoid.

Further there exist inverse b for every element. $a \oplus b = 0$ gives $b= \dfrac{-a}{1+a}$ Observe $a$ can't be $-1$ and it is mentioned in question, therefore inverse will always exist. Therefore it is group.

Finally, here $\oplus$ is commutative. Therefore it is an abelian group.