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\begin{document}

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\section*{equivalence relation}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{relations}{}\paragraph*{{Relations}}\label{relations}

[[!include relations - contents]]

\hypertarget{equivalence_relations}{}\section*{{Equivalence relations}}\label{equivalence_relations}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{setoids}{Setoids}\dotfill \pageref*{setoids} \linebreak
\noindent\hyperlink{variants_and_generalizations}{Variants and generalizations}\dotfill \pageref*{variants_and_generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An \textbf{equivalence relation} on a set $S$ is a binary [[relation]] $\equiv$ on $S$ that is:

\begin{itemize}%
\item [[reflexive relation|reflexive]]: $x \equiv x$ for all $x: S$;
\item [[symmetric relation|symmetric]]: $x \equiv y$ if $y \equiv x$; and
\item [[transitive relation|transitive]]: $x \equiv z$ if $x \equiv y \equiv z$.

\end{itemize}
(One can also define it as a relation that is both reflexive and [[euclidean relation|euclidean]].) The [[de Morgan duality|de Morgan dual]] of an equivalence relation is an [[apartness relation]].

If $R$ is any relation on $S$, there is a smallest equivalence relation containing $S$ (noting that an intersection of an arbitrary family of equivalence relations is again such; see also [[Moore closure]]), called the equivalence relation \emph{generated by} $R$.

\hypertarget{setoids}{}\subsection*{{Setoids}}\label{setoids}

A \textbf{setoid} is a set equipped with an equivalence relation. (However, we should be cautious with this terminology, since the people who typically use setoids begin with an impoverished notion of set and then introduce setoids specifically to fix this, as described below.)

Equivalently, a setoid is a [[groupoid]] [[enriched category|enriched]] over the [[cartesian monoidal category]] of [[truth values]]. Equivalently, a setoid is a [[groupoid]] that is [[0-truncated]]. Then the equivalence relation on $S$ is a way of making $S$ into the set of objects of such a groupoid. Equivalently, a setoid is a [[(0,1)-category]] whose each morphism is iso, or a [[symmetric relation|symmetric]] [[preordered set]].

It may well be useful to consider several possible equivalence relations on a given set. When considering a single equivalence relation once and for all, however, it is common to take the [[quotient set]] $S/{\equiv}$ and use that instead. As a groupoid, any setoid is [[equivalence of categories|equivalent]] to a [[set]] in this way (although in the absence of the [[axiom of choice]], it is only a ``weak'' or [[anafunctor|ana-equivalence]]).

Setoids are still important in [[foundations]] of mathematics where quotient sets don't always exist and the above equivalence cannot be carried out. However, arguably this is a terminological mismatch, and such people should say `set' where they say `setoid' and something else (such as `[[preset]]', `[[type]]', or `[[completely presented set]]') where they say `set'. (See [[Bishop set]] and page 9 of \href{http://www.cs.chalmers.se/Cs/Research/Logic/TypesSS05/Extra/palmgren.pdf}{these lecture notes}.)

\hypertarget{variants_and_generalizations}{}\subsection*{{Variants and generalizations}}\label{variants_and_generalizations}

\begin{itemize}%
\item A \emph{[[partial equivalence relation]]} is a symmetric and transitive relation.


\item A \emph{[[congruence]]} is a notion of equivalence relation [[internalization|internal]] to a suitable [[category]].


\item The generalization of this to [[(∞,1)-category theory]] is that of \emph{[[groupoid object in an (∞,1)-category]]}.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Bishop set]]


\item [[congruence]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

For the history of the notion of equivalence relation see \emph{\href{http://mathoverflow.net/questions/135347/who-introduced-the-terms-equivalence-relation-and-equivalence-class}{this MO discussion}}.

[[!redirects equivalence relation]] [[!redirects equivalence relations]]

[[!redirects setoid]] [[!redirects setoids]]



\end{document}