# Let $\ast$ be a binary operation on $R-\left \{ -1 \right \}$ defined by $a\ast b=\frac{a}{b+1}$ for all $a,b \in R-\left \{ -1 \right \}$Show that $\ast$ is neither commutative nor associative in $R-\left \{ -1 \right \}$.

$a\ast b=\frac{a}{b+1}$ for all $a,b \in R-\left \{ -1 \right \}$

(i) eg: $2.3=\frac{2}{3+1}=\frac{2}{4}=\frac{1}{2}$

$3.2=\frac{3}{2+1}= \frac{3}{3}=1$

$\because (a\ast b)\neq (b\ast a)$

So, it is not commutative.

(ii) associative

eg: $(2.3).1=(\frac{2}{3+1}).1=(\frac{1}{2}).1$

$\Rightarrow \frac{1/2}{1+1}=\frac{1/2}{2}= \frac{1}{4}$

$2.(3.1)\Rightarrow 2.\left ( \frac{3}{1+1} \right )\Rightarrow \frac{3}{2}\Rightarrow$

$\Rightarrow \frac{2}{\frac{3}{2}+1}\Rightarrow \frac{2}{\frac{3+2}{2}}\Rightarrow \frac{2}{5}\times \frac{1}{2}= \frac{1}{5}$

$\left ( a\ast b \right )\ast c\neq a\ast \left ( b\ast c \right )$

So, it is not associative.

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