Move `OrderedPair` and `Relation` to Enderton.

These modules probably won't be very useful in general; there exist
better representations in Lean when dealing with ordered pairs or
relations already.
finite-set-exercises
Joshua Potter 2023-06-29 15:25:59 -06:00
parent c65e28888d
commit dd4340c4bd
7 changed files with 51 additions and 13 deletions

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@ -1,3 +1,5 @@
import Bookshelf.Enderton.Set.Chapter_1 import Bookshelf.Enderton.Set.Chapter_1
import Bookshelf.Enderton.Set.Chapter_2 import Bookshelf.Enderton.Set.Chapter_2
import Bookshelf.Enderton.Set.Chapter_3 import Bookshelf.Enderton.Set.Chapter_3
import Bookshelf.Enderton.Set.OrderedPair
import Bookshelf.Enderton.Set.Relation

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@ -73,6 +73,12 @@ There is a set having no members:
\end{axiom} \end{axiom}
\section{\partial{Equivalence Relation}}%
\label{ref:equivalence-relation}
Relation $R$ is an \textbf{equivalence relation} if and only if $R$ is a binary
relation that is reflexive, symmetric, and transitive.
\section{\defined{Extensionality Axiom}}% \section{\defined{Extensionality Axiom}}%
\label{ref:extensionality-axiom} \label{ref:extensionality-axiom}
@ -236,6 +242,12 @@ Given \nameref{ref:relation} $R$, the \textbf{range} of $R$, denoted $\ran{R}$,
\end{definition} \end{definition}
\section{\partial{Reflexive Relation}}%
\label{ref:reflexive-relation}
A binary relation $R$ is \textbf{reflexive} on $A$ if and only if $xRx$ for all
$x \in A$.
\section{\defined{Relation}}% \section{\defined{Relation}}%
\label{ref:relation} \label{ref:relation}
@ -272,6 +284,12 @@ For each formula $\phi$ not containing $B$, the following is an axiom:
\end{axiom} \end{axiom}
\section{\partial{Symmetric Relation}}%
\label{ref:symmetric-relation}
A binary relation $R$ is \textbf{symmetric} on $A$ if and only if whenever
$xRy$ then $yRx$.
\section{\defined{Symmetric Difference}}% \section{\defined{Symmetric Difference}}%
\label{ref:symmetric-difference} \label{ref:symmetric-difference}
@ -284,6 +302,12 @@ The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set
\end{definition} \end{definition}
\section{\partial{Transitive Relation}}%
\label{ref:transitive-relation}
A binary relation $R$ is \textbf{transitive} on $A$ if and only if whenever
$xRy$ and $yRz$, then $xRz$.
\section{\defined{Union Axiom}}% \section{\defined{Union Axiom}}%
\label{ref:union-axiom} \label{ref:union-axiom}
@ -3148,6 +3172,25 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\section{Equivalence Relations}%
\label{sec:equivalence-relations}
\subsection{\unverified{Theorem 3M}}%
\label{sub:theorem-3m}
\begin{theorem}[3M]
If $R$ is a symmetric and transitive relation, then $R$ is an equivalence
relation on $\fld{R}$.
\end{theorem}
\begin{proof}
TODO
\end{proof}
\section{Exercises 3}% \section{Exercises 3}%
\label{sec:exercises-3} \label{sec:exercises-3}

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@ -1,9 +1,6 @@
import Mathlib.Data.Set.Basic
import Mathlib.Data.Set.Lattice
import Bookshelf.Enderton.Set.Chapter_1 import Bookshelf.Enderton.Set.Chapter_1
import Common.Logic.Basic
import Common.Set.Basic import Common.Set.Basic
import Mathlib.Data.Set.Lattice
/-! # Enderton.Chapter_2 /-! # Enderton.Chapter_2

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@ -1,6 +1,6 @@
import Bookshelf.Enderton.Set.Chapter_2 import Bookshelf.Enderton.Set.Chapter_2
import Common.Set.OrderedPair import Bookshelf.Enderton.Set.OrderedPair
import Common.Set.Relation import Bookshelf.Enderton.Set.Relation
/-! # Enderton.Chapter_3 /-! # Enderton.Chapter_3

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@ -1,6 +1,4 @@
import Common.Set.OrderedPair import Bookshelf.Enderton.Set.OrderedPair
import Mathlib.Data.Set.Basic
import Mathlib.Data.Set.Prod
/-! # Relations /-! # Relations

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@ -1,5 +1,3 @@
import Common.Set.Basic import Common.Set.Basic
import Common.Set.Interval import Common.Set.Interval
import Common.Set.OrderedPair
import Common.Set.Partition import Common.Set.Partition
import Common.Set.Relation