Question

Examine whether the operation $\ast$ defined on $R$ by $a\ast b=ab+1$ is
(i) a binary or not.
(ii) if a binary operation, is it associative or not ?

Answer 1

$\text{Given,}a\ast b=ab+1$
(i) Since $a,b\in R,ab$ will belong to $R$
$\text{i.e.,}ab\in R$
$\Rightarrow ab+1\in R$
$\text{So,}a\ast b$ is a binary operation.

(ii) The binary operation will be associative if
$(a\ast b)\ast c=a\ast (b\ast c)\forall a,b,c\in R$
$\text{Consider}(a\ast b)\ast c=(ab+1)\ast c$
$=(ab+1)c+1=abc+c+1$
$a\ast (b\ast c)=a\ast (bc+1)$
$=a(bc+1)+1=abc+a+1$
Here, we can see that
$(a\ast b)\ast c\ne a\ast (b\ast c)\forall a,b,c\in R$
So, it is not associative.