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 \}.

 

 

 

 
 
 
 
 

Answers (1)

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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