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# Find which of the operations given below has identity: (i) $a \ast b = a - b$.

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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 $a \ast b = a - b$
let e be an identity element of * operation
By definition of identity element a*e=e*a =a
But by definition of * operation a*e =a-e=> a=a-e => e=-1
and e*a=e-a =>a= e-a => e= 2a
we get two different values for e the identity element
This is not possible as the identity is unique. Therefore the operation defined by $a \ast b = a - b$ has no identity element.
answered Mar 20, 2013