# Find which of the operations given below has identity: $(iv)\;\; a \ast b = (a-b)^2$

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Toolbox:
• An element $e \in N$ is an identify element for operation * if $a \ast e=e \ast a$ for all $a \in N$
Given $a *b =(a-b)^2$
let e be an identity element of the operation*
We know that $a \ast e = a, a \neq 0 \$
by definition of * operation $a*e=(a-e)^2$ =>$a=(a-e)^2$
=> a-e = square root a , since square root of a does not beleng to N for all a in N , the identity element e is not defined
Therefore and the operation* defined by$a *b =(a-b)^2$ for all $a \in N$ has no identity.
answered Mar 20, 2013