Question

Let z be the set of integers and o be a binary operation on z defined as $aob = a + b - ab$ for all $a,b \in z$. The inverse of an element $a( \neq 1) \in z$ is

• A. $\frac{a}{a - 1}$
• B. $\frac{a}{1 - a}$
• C. $\frac{a - 1}{a}$
• D. None of these

Answer 1

Answer: (A)

$\frac{a}{a - 1}$

Answer 2

Let e be the identity element for the binary operation o defined on z given by $aob = a + b - ab$

Then $aoe = a = eoa$ for all $a \in z$

⇒ $a + e - ae = a$ for all $a \in z$ ⇒ $e(1 - a) = 0$ for all $a \in z$ ⇒ $e = 0$.

So, 0 is the identity element for the binary operation o and z.

Let x be the inverse of $a \in z$. Then, $aox = xoa = 0$

⇒ $a + x - ax = 0$ ⇒ $x(1 - a) = - a$ ⇒ $x = \frac{a}{a - 1}$ $(\because a \neq 1)$

Thus, $\frac{a}{a - 1}$ is the inverse of $a( \neq 1) \in z$