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.