Question

Find the identity element in the set $Q^{+}$ of all positive rational numbers for the operation $\ast$ defined by $a\ast b=\frac{ab}{2}$ for all $a,b\in Q^{+}$.

• A. $\varphi$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is D $2$

Let e be the identity element in $Q^{+}$. Then,

$a\ast e=a=e\ast a$ for all $a\in Q^{+}$

$a\ast e=a$ and $e\ast a=a$ for all $a\in Q^{+}$

$\frac{ae}{2}=a$ and $\frac{ea}{2}=a$ for all $a\in Q^{+}$

$e=2$

Hence, 2 is the identity element in $Q^{+}$.