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