Question

Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b −ab. Then, the identify element for * is
(a) 1

(b) $\frac{a-1}{a}$

(c) $\frac{a}{a-1}$

(d) 0

Answer 1

(d) 0

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

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

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