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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{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