Show that * on R –{ - 1}, defined by


Show that $*$ on $R-\{-1\}$, defined by $(a * b)=\frac{a}{(b+1)}$ is neither commutative nor associative.


let $a=1, b=0 \in R-\{-1\}$

$a^{*} b=\frac{1}{0+1}=1$

And $b^{*} a=\frac{0}{1+1}=0$

Hence * is not commutative.

Let c = 3.

$(a * b) * c=1 * c=\frac{1}{3+1}=\frac{1}{4}$

$a *(b * c)=a * \frac{0}{3+1}=1 * 0=\frac{1}{0+1}=1$

Hence * is not associative.


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