Question

Binary operation $\ast$ on $R-\{-1\}$ defined by $a\ast b=\frac{a}{b+1}$ is

• A. $\ast$ is associative and commutative
• B. $\ast$ is associative but not commutative
• C. $\ast$ is neither associative nor commutative
• D. $\ast$ is commutative but not associative

Answer 1

The correct option is C $\ast$ is neither associative nor commutative

$a\ast b=\frac{a}{b+1}$
$b\ast a=\frac{b}{a+1}$
clearly, $a\ast b\ne b\ast a$. So, $\ast$ is not commutative

$a\ast (b\ast c)=\frac{a}{b\ast c+1}=\frac{a}{\frac{b}{c+1}+1}=\frac{a(c+1)}{b+c+1}$
$(a\ast b)\ast c=\frac{a\ast b}{c+1}=\frac{\frac{a}{b+1}}{c+1}=\frac{a}{(b+1)(c+1)}$
Clearly, $a\ast (b\ast c)\ne (a\ast b)\ast c$.So, $\ast$ is not associtive