From 442d732696ad431b84f6e5c72b6ee785be4fd968 Mon Sep 17 00:00:00 2001
From: adelon <22380201+adelon@users.noreply.github.com>
Date: Sat, 10 Feb 2024 02:22:14 +0100
Subject: Initial commit

---
 library/relation/closure.tex | 28 ++++++++++++++++++++++++++++
 1 file changed, 28 insertions(+)
 create mode 100644 library/relation/closure.tex

(limited to 'library/relation/closure.tex')

diff --git a/library/relation/closure.tex b/library/relation/closure.tex
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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
-- 
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