Question

Let $\ast$ be a operation defined on the set of rational numbers by $a\ast b=\frac{ab}{4}$, find the identity element.

Answer 1

$\ast$ is defined by $a\ast b=\frac{ab}{4}$, where $a,b\in Q$ ........... $\left(1\right)$

$e$ is the identity element of $\ast$ defined on $Q$ if

$a\ast e=a=e\ast a$

$⟹\frac{ae}{4}=a$ ........... By $\left(1\right)$

$⟹e=4$

Similarly, if we take $e\ast a=a$ we get $e=4$

$\therefore$ $e=4$ is the identity element.