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Question
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .
Options
a
`1/a`
`2/a`
`4/a`
Solution
`4/a`
Let e be the identity element in Q+ with respect to * such that
\[a * e = a = e * a, \forall a \in Q^+ \]
\[a * e = a \text{ and } e * a = a, \forall a \in Q^+ \]
\[\frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in Q^+ \]
\[e = 2 \in Q^+ , \forall a \in Q^+\]
Thus, 2 is the identity element in Q+ with respect to *.
\[\text{ Let }a \in Q^+ \text{ and }b \in Q^+ \text{ be the inverse of a } . \]
\[\text{ Then },\]
\[a * b = e = b * a\]
\[a * b = e \text{ and } b * a = e\]
\[\frac{ab}{2} = 2 \text{ and }\frac{ba}{2} = 2\]
\[b = \frac{4}{a} \in Q^+ \]
\[\text{ Thus },\frac{4}{a}\text{ is the inverse of a } \in Q^+ . \]
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