Question

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by  \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\]:

Find the invertible elements of Q_{0 ?}

Answer 1

\[\text{ Let }a \in Q_0 \text{ and }b \in Q_0 \text{ be the inverse of a } . \text{Then},\] 
\[a o b = e = b o a\] 
\[ \Rightarrow a o b = e \text{ and }b o a = e\] 
\[ \Rightarrow \frac{ab}{2} = 2 \text{ and }\frac{ba}{2} = 2\] 
\[ \Rightarrow b = \frac{4}{a} \in Q_0 \] 
\[\text{Thus},\frac{4}{a} \text{is the inverse of a} \in Q_0 . \]