Q ^ + is the set of all positive rational numbers with the binary… - Vidyadip

MCQ

$Q ^{+}$is the set of all positive rational numbers with the binary operation * defined by $a^ * b=\frac{ ab }{2} \forall a , b \in Q ^{+}$. The inverse of an element $a \in Q ^{+}$ is:

A
$\text{a}$
B
$\frac{1}{\text{a}}$
C
$\frac{2}{\text{a}}$
✓
$\frac{4}{\text{a}}$

✓

Answer

Correct option: D.

$\frac{4}{\text{a}}$

Let e be the identity element in $Q^+$ with respect to $*$ such that
$a^ * e = a = e^ * a, \forall\text{ a}\in\text{Q}^+$
$a^ * e = a$ and $e^ * a = a, \forall\text{ a}\in\text{Q}^+$
$\frac{\text{ae}}2=\text{a}$ and $\frac{\text{ea}}2=\text{a}$, $\forall\text{ a}\in\text{Q}^+$
$\text{e}=2\in\text{Q}^+, \forall\text{ a}\in\text{Q}^+$
Thus$, 2$ is the identity element in $Q^+$ with respect to $*.$
Let $\text{ a}\in\text{Q}^+$ and $\text{ b}\in\text{Q}^+$ be the inverse of $a.$
Then,
$a^ * e = a = e^ * a$
$a^ * b = e$ and $b^ * a = e$
$\frac{\text{ab}}2=2$ and $\frac{\text{ba}}2=2$
$\text{b}=\frac{4}{\text{a}}\in\text{Q}^+$
Thus, $\frac{4}{\text{a}}$ is the inverse of $\text{ a}\in\text{Q}^+$.

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