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Revision difference : eventually coincide |
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Let $A$ and $B$ be two nonempty sets of integers. We say that $A$ and $B$ \emph{eventually coincide} if there is an integer $C$ such that $n\in A$ if and only if $n\in B$ for all $n\geq C$. In this case, we write $A\sim B$, noting that the relation of eventually coinciding is clearly an equivalence relation. While a seemingly trivial notation, this turns out to be the ``right'' notion of \PMlinkescapetext{equivalence} of sets when dealing with asymptotic properties such as \PMlinkescapetext{density}.
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Let $A$ and $B$ be two nonempty sets of integers. We say that $A$ and $B$ \emph{eventually coincide} if there is an integer $C$ such that $n\in A$ if and only if $n\in B$ for all $n\geq C$. In this case, we write $A\sim B$, noting that the relation of eventually coinciding is clearly an equivalence relation. While a seemingly trivial notation, this turns out to be the ``right'' notion of equivalence of sets when dealing with asymptotic properties such as density.
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{Nathanson} |
\bibitem{Nathanson} |
| Nathanson, Melvyn B., \emph{Elementary Methods in Number Theory}, Graduate Texts in Mathematics, Volume 195. Springer-Verlag, 2000. |
Nathanson, Melvyn B., \emph{Elementary Methods in Number Theory}, Graduate Texts in Mathematics, Volume 195. Springer-Verlag, 2000. |
| \end{thebibliography} |
\end{thebibliography} |
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