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# Find which of the operations given below has identity:$(v)\;\; a \ast b = \large\frac {ab} {4}$

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• An element $e \in N$ is an identify element for operation * if $a \ast e=e \ast a$ for all $a \in N$
Given $a \ast b = \large\frac {ab} {4}$
Let e be the identity element of the * operation
then from definition of identity element a*e=e*a=a
from definition of * operation $a*e= \large\frac {ae} {4}$ =>$a= \large\frac {ae} {4}$ =.> e=4
Also$e*a= \large\frac {ea} {4}$ =>$a=\large\frac {ea} {4}$ => e=4
e =4 is the identity element,
solution - the given operation *given by $a \ast b = \large\frac {ab} {4}$ has a n identity element