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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 Q0 ?
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Solution
\[\text{ Let }a \in Q_0 \text{ and }b \in Q_0 \text{ be the inverse of a } . \text{Then},\]
\[a o b = e = b o a\]
\[ \Rightarrow a o b = e \text{ and }b o a = e\]
\[ \Rightarrow \frac{ab}{2} = 2 \text{ and }\frac{ba}{2} = 2\]
\[ \Rightarrow b = \frac{4}{a} \in Q_0 \]
\[\text{Thus},\frac{4}{a} \text{is the inverse of a} \in Q_0 . \]
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