Question

Let z be the set of integers and 0 be binary operation of z defined as a 0 b = a + b - ab for all a, b $\in$ z. The inverse of an element a( $\ne$1) $\in$ z is:

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

Answer 1

Answer: (A)

Let e be the identity element. And we have a .e = a $\Rightarrow$ a + e - ae = a $\Rightarrow$ e - ae = 0 $\Rightarrow$ either e = 0 or 1 - a = 0 but a $\ne$ 1.
Thus e = 0 is the identity.
We know that, $a{a}^{-1}$ = e;
Now let A is the inverse of a. Thus a . A = 0
$\Rightarrow$ a + A - a A = 0 $\Rightarrow$ A = $\frac{a}{a-1}\in$ Q thus A is the founded inverse of a.