Question

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?

Answer 1

Let e be the identity element in Q\[-\]{\[-\]1} with respect to * such that

\[a * e = a = e * a, \forall a \in Q - \left\{ - 1 \right\}\] 
\[ \Rightarrow a * e = a \text{ and }e * a = a, \forall a \in Q - \left\{ - 1 \right\}\] 
\[ \Rightarrow a + e + ae = a \text{ and } e + a + ea = a, \forall a \in Q - \left\{ - 1 \right\}\] 
\[ \Rightarrow e\left( 1 + a \right) = 0, \forall a \in Q - \left\{ - 1 \right\}\] 
\[ \Rightarrow e = 0, \forall a \in Q - \left\{ - 1 \right\} \left[ \because a\neq-1 \right]\] 

Thus, 0 is the identity element inQ\[-\]{\[-\]1} with respect to *.