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Q.
For binary operation $*$ defined on $R - \{1\}$ such that $a * b=\frac{a}{b+1}$ is
Relations and Functions - Part 2
Solution:
Commutative : $a * b$ $=\frac{a}{b+1}$ and $b * a=\frac{b}{a+1}$
$a * b \ne b * a$
$\Rightarrow *$ is not commutative. Associative:
$(a*b)*c =\left(\frac{a}{b+1}\right)*c=\frac{\left(\frac{a}{b+1}\right)}{c+1}$
$=\frac{a}{\left(b+1\right)\left(c+1\right)}$
and $a*\left(b*c\right)=a* \left(\frac{b}{c+1}\right)$
$=\frac{a}{\left(\frac{b}{c+1}\right)+1}$
$=\frac{a\left(c+1\right)}{b+c+1}$
So, $(a * b) * c \ne a * (b * c)$.
Hence, $*$ is not associative.