\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
