Question

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

• A. * is associative and cummutative
• B. * is neither associative nor commutative
• C. * is commutative but not associative
• D. * is associative but not commutative

Answer 1

The correct option is B * is neither associative nor commutative

$a\ast b=\frac{a}{b+1}$
$b\ast a=\frac{b}{a+1}$
$\frac{a}{b+1}\ne \frac{b}{a+1}\Rightarrow a\ast b\ne b\ast a$
$\therefore \ast$ is not commutative
$a\ast (b\ast c)=a\ast \left[\frac{b}{c+1}\right]=\frac{a}{\frac{b}{c+1}+1}$
$=\frac{a(c+1)}{b+c+1}$
$(a\ast b)\ast c=\left(\frac{a}{b+1}\right)\ast c=\frac{\frac{a}{b+1}}{c+1}$
$=\frac{a}{(b+1)(c+1)}$
$a\ast (b\ast c)\ne (a\ast b)\ast c$
$\therefore \ast$ is neither associative nor commutative