Question

A binary operation $\ast$ defined $Q-\left\{1\right\}$ is given by $a\ast b=a+b-ab$. Find the identity element.

Answer 1

For a binary operation, if $a\times e=a=e\times a$,then $‘e^{\prime }$ is know as the identity of the operator.

Now, $a\times b=a+b-ab$

Right identity

$a\times e=a+e-ae=a$

$e-ae=0$

$e=\frac{0}{1-a}=0$, for all $a\ne 1$, as domain is $Q-\left\{1\right\}$

Let identity

$e\times b=e+b-eb=b$

$e-eb=0$

$e(1-b)=0$

$e=\frac{0}{1-b}=0$

As the left and right identity is same. So, the identity of operation is $‘0^{\prime }$.