Here is a very tricky question that I am trying to solve since last three days but until now I couldn't find the solution. Can anyone be of some help to solve this question?

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Does a binary operation

exist such that

and

are isomorphic? Prove your answer.

NOTE:

is the set of natural numbers,

is the set of integers and + is the addition operation. Two algebraic structures

and

are isomorphic if and only if a bijection

exists such that

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I would appreciate if full proof of this problem can be written

Bundle of thanks in advance.

Regards