Question

Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is
(a) 0
(b) 1
(c) $\frac{1}{2}$
(d) −1

Answer 1

(a) 0

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

$$\begin{gathered} a*e=a=e*a,\forall a\in Q-\left\{1\right\} \\ a*e=a\text{and}e*a=a,\forall a\in Q-\left\{1\right\} \\ \mathrm{Then}, \\ a+e-ae=a\text{and}e+a-ea=a,\forall a\in Q-\left\{1\right\} \\ e\left(1-a\right)=0,\forall a\in Q-\left\{1\right\} \\ e=0\in Q-\left\{1\right\}\left[\text{∵}a\ne 1\right] \end{gathered}$$

Thus, 0 is the identity element in Q $-$ {1} with respect to *.