Question

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

• A. $\frac{a}{16}$
• B. $16a$
• 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,
$\Rightarrow a\ast e=a=e\ast a,\forall a\in Q_0$
$\Rightarrow a\ast e=a$
$\Rightarrow \frac{ae}{4}=a$
$e=4$
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, $a\ast b=e=b\ast a$
$\Rightarrow \frac{ab}{4}=4=\frac{ba}{4}$
$\Rightarrow ab=16$
$\Rightarrow b=\frac{16}{a}$
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 $\frac{16}{a}$.