Consider a binary operation on $\mathrm{Q}-\{1\}$, defined by $\mathrm{a} * \mathrm{~b}=\mathrm{a}+\mathrm{b}-\mathrm{ab}$.
(i) Find the identity element in $\mathrm{Q}-\{1\}$.
(ii) Show that each $a \in Q-\{1\}$ has its inverse.
(i) For a binary operation $*$, e identity element exists if $a * e=e^{*} a=a .$ As $a * b=a+b-a b$
$a * e=a+e-a e(1)$
$e^{*} a=e+a-e a(2)$
using $a^{*} e=a$
$a+e-a e=a$
$e-a e=0$
$e(1-a)=0$
either $e=0$ or $a=1$ as operation is on $Q$ excluding 1 so $a \neq 1$, hence $e=0$.
So identity element $e=0$.
(ii) for a binary operation $*$ if e is identity element then it is invertible with respect to $*$ if for an element $b, a^{*} b=e=b^{*} a$ where $b$ is called inverse of $*$ and denoted by $a^{-1}$.
$a * b=0$
$a+b-a b=0$
$b(1-a)=-a$
$\mathrm{b}=\frac{-\mathrm{a}}{(1-\mathrm{a})} \Rightarrow \frac{\mathrm{a}}{(\mathrm{a}-1)}$
$a^{-1}=\frac{a}{(a-1)}$