Question

Find if the given below operation has identity?

(iv) $a\ast b=(a-b{)}^2$

Answer 1

Given: $a\ast b=(a-b{)}^2$

$e$ is the identity of $\ast$ if

$a\ast e=e\ast a=ai.e.,(a-e{)}^2=(e-a{)}^2=a$

There is no possible value of $e$ where,

$(a-e{)}^2=a$

So, $\ast$ has no identity element in $Q.$