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`

Answer 1

`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 *.

\[\text{ Let }a \in Q^+ \text{ and }b \in Q^+ \text{ be the inverse of a } . \]
\[\text{ Then },\]
\[a * b = e = b * a\]
\[a * b = e \text{ and } b * a = e\]
\[\frac{ab}{2} = 2 \text{ and }\frac{ba}{2} = 2\]
\[b = \frac{4}{a} \in Q^+ \]
\[\text{ Thus },\frac{4}{a}\text{ is the inverse of a } \in Q^+ . \]