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Does a binary operation * exist such that (N;+) and (Z;*) are isomorphic? N : set of natural numbers, Z : set of integers and + is the addition operation. If yes, please give me an isomorphism function?

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1 Answer 1

Whenever you have two sets that are in bijection you can "transport" an operation, or any other algebraic structure, from one to the other. Namely, if $A$ is set endowed with an operation $+$ and $f:A\rightarrow B$ is a bijection of sets, you can define an operation $\ast$ on $B$ simply as $$ b\ast b^\prime=f(a+a^\prime) $$ where $a$ and $a^\prime$ are the unique elements in $A$ such that $f(a)=b$ and $f(a^\prime)=b^\prime$. Then $(B,\ast)$ is isomoprphic to $(A,+)$ by construction.

This can be applied to the question's case since there are surely bijections ${\Bbb N}\rightarrow{\Bbb Z}$. The problem is that the operations on $\Bbb Z$ that you construct in this way are very artificial and quite uninteresting.

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