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 ?

Answer 1

\[\text{ Let }a \in Q - \left\{ - 1 \right\} \text{ and } b \in Q - \left\{ - 1 \right\} \text{be the inverse of a} . \text{ Then }, \] 
\[a * b = e = b * a\] 
\[ \Rightarrow a * b = e \text{ and }b * a = e\] 
\[ \Rightarrow a + b + ab = 0 \text{ and }b + a + ba = 0\] 
\[ \Rightarrow b\left( 1 + a \right) = - a \in Q - \left\{ - 1 \right\}\] 
\[ \Rightarrow b = \frac{- a}{1 + a} \in Q - \left\{ - 1 \right\} \left[ \because a \neq - 1 \right]\] 
\[\text{Thus},\frac{- a}{1 + a} \text{is the inverse of a} \in Q - \left\{ - 1 \right\} . \]