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Question
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
Solution
\[\text{ Let }a \in Q - \left\{ - 1 \right\} \text{ and } b \in Q - \left\{ - 1 \right\} \text{be the inverse of a} . \text{ Then }, \]
\[a * b = e = b * a\]
\[ \Rightarrow a * b = e \text{ and }b * a = e\]
\[ \Rightarrow a + b + ab = 0 \text{ and }b + a + ba = 0\]
\[ \Rightarrow b\left( 1 + a \right) = - a \in Q - \left\{ - 1 \right\}\]
\[ \Rightarrow b = \frac{- a}{1 + a} \in Q - \left\{ - 1 \right\} \left[ \because a \neq - 1 \right]\]
\[\text{Thus},\frac{- a}{1 + a} \text{is the inverse of a} \in Q - \left\{ - 1 \right\} . \]
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