# Isomorphic of (N,+)?

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?

-

## 1 Answer

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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 14, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9616856575012207, "perplexity_flag": "head"}