Question

Let $*$ be a binary operation on $Q_0$ (the set of all non-zero rational numbers) defined by $a * b=\frac{a b}{4}$, $a, b \in Q_0$ Then, the inverse of an element in $Q_0$ is

• A. $\frac{a}{16}$
• B. $16 a$
• C. $\frac{16}{a}$
• D. Does not exist

Answer 1

Answer: (C)

To find the inverse of an element in $Q_0$, we first find the identity element in $Q_0$.
Now, lete be the identity element in $Q_0$. Then,
$ \Rightarrow a * e=a=e * a, \forall a \in Q_0 $
$ \Rightarrow a * e=a $
$ \Rightarrow \frac{a e}{4}=a$
$ e=4$
Thus, 4 is the identity element in $Q_0$ for the binary operation
Now, let a be an invertible element in $Q_0$ and $b$ be its inverse.
Then, $ a * b =e=b * a $
$\Rightarrow \frac{a b}{4} =4=\frac{b a}{4}$
$\Rightarrow a b =16 $
$\Rightarrow b=\frac{16}{a} $
Clearly, $ \frac{16}{a} \in Q_0, \forall a \in Q_0$
$\therefore$ Every element of $Q_0$ is invertible and the inverse of an element $a \in Q_0$ is $\frac{16}{a}$.