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\import{relation/properties.tex}
\import{relation/equivalence.tex}
\subsection{Closure operations on relations}
\begin{definition}\label{reflexive_closure}
$\reflexiveClosure{X}{R} = R\union\identity{X}$.
\end{definition}
% reflexive closure of R is the smallest reflexive relation containing R
\begin{proposition}\label{reflexive_closure_is_reflexive}
$\reflexiveClosure{X}{R}$ is reflexive on $X$.
\end{proposition}
\begin{proof}
Follows by \cref{reflexive_closure,reflexive_on,union_iff,id_iff}.
\end{proof}
\begin{definition}\label{reflexive_reduction}
$\reflexiveReduction{X}{R} = R\setminus\identity{X}$.
\end{definition}
\begin{definition}\label{symmetric_closure}
$\symmetricClosure{R} = R\union\converse{R}$.
\end{definition}
% LATER transitive closure
% LATER reflexive transitive closure
% LATER equivalence closure
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