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

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

MCQ

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