Question

The operation* is defined on the set Q of rational numbers by a*b= ab+a+b,a,b belongs
to Q,then find the element of Q which does not have an inverse under*

Answer 1

Question 1.

The operation $*$ is defined on the set $\mathbb{Q}$ of rational numbers by $a*b=ab+a+b$, $a,b\in\mathbb{Q}$. Find the element of $\mathbb{Q}$ which does not have an inverse under $*$.

Solution. First of all note that $*$ is commutative and is the identity under $*$. Indeed,

$a*0=0*a=a\cdot 0+a+0=a.$

Now take $a\in\mathbb{Q}$ and suppose there is $b\in\mathbb{Q}$ such that

$a*b=0\Leftrightarrow ab+a+b=0\Leftrightarrow b(a-1)=-a.$

We see that if $a=1$, then this equality becomes $0=-1$, which is a contradiction. So, $a=1$ is not invertible. But if $a\neq 1$, then $b=\frac{a}{1-a}\in\mathbb{Q}$ is the inverse of $a$.

Answer: 1. $\Box$