Question

Let ${}^{\prime }{\ast }^{\prime }$ be a binary operation on set Q of rational number defined as $a\ast b=\frac{ab}{5}$. Write the identity for ${}^{\prime }{\ast }^{\prime }$, if any.

Answer 1

Given, binary operation is $a\ast b=\frac{ab}{5}$

Now, let e be the identity element of ${}^{\prime }{\ast }^{\prime }$ on Q.

Then,

$a\ast e=a=e\ast a$ for every $a\in Q$

$\frac{ae}{5}=a=\frac{ea}{5}$

$ae=5a⟹e=5,a\ne 0$

Also, for $0\in Q$, we have,

$0\ast 5=0=5\ast 0$

Hence, $e=5$ is the identity element.