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Question
Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .
Options
`4/3`
2
`1/3`
`2/3`
Solution
`4/3`
Let e be the identity element in Q+ with respect to \[\odot\] such that
\[a * e = a = e * a, \forall a \in Q^+ \]
\[a * e = a \text{ and }e * a = a, \forall a \in Q^+ \]
\[\frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in Q^+ \]
\[e = 2 , \forall a \in Q^+\]
Thus, 2 is the identity element in Q+ with respect to \[\odot.\]
\[\text{ Let }b \in Q^+ \text{ be the inverse of 3 . Then},\]
\[3 * b = e = b * 3\]
\[3 * b = e \text {and }b * 3 = e\]
\[\frac{3b}{2} = 2 \text { and }\frac{b\left( 3 \right)}{2} = 2\]
\[b = \frac{4}{3}\]
\[\text {Thus},\frac{4}{3} \text{ is the inverse of 3 } . \]
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