From 140195a8ee29a0a007e806900d859a2fb7da32c2 Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Sun, 2 Jul 2023 12:26:09 -0600 Subject: [PATCH] Enderton. Add equivalence relation problem prompts. --- Bookshelf/Enderton/Set.tex | 213 +++++++++++++++++++++++++- Bookshelf/Enderton/Set/Chapter_3.lean | 3 +- 2 files changed, 214 insertions(+), 2 deletions(-) diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index 8a95919..c24833a 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -32,6 +32,13 @@ For any relation $R$ there is a function $H \subseteq R$ with For any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$ for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$ +\section{\pending{Compatible}}% +\label{sec:compatible} + +A \nameref{ref:function} $F$ is \textbf{compatible} with relation $R$ if and + only if for all $x$ and $y$ in $A$, + $$xRy \Rightarrow F(x)RF(y).$$ + \section{\defined{Composition}}% \label{ref:composition} @@ -3269,7 +3276,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{proof} -\subsection{\sorry{Theorem 3P}}% +\subsection{\pending{Theorem 3P}}% \label{sub:theorem-3p} \begin{theorem}[3P] @@ -3280,6 +3287,50 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one. \end{theorem} +\begin{proof} + + Let $\Pi = \{[x]_R \mid x \in A\}$. + We show that (i) no two different sets in $\Pi$ have any common elements and + (ii) that each element of $A$ is in some set in $\Pi$. + + \paragraph{(i)}% + + Let $[x]_R, [y]_R \in \Pi$ be two different sets. + We must show that $[x]_R \cap [y]_R = \emptyset$. + For the sake of contradiction, suppose $[x]_R \cap [y]_R \neq \emptyset$. + Let $z \in [x]_R \cap [y]_R$. + Then $xRz$ and $yRz$. + Since $R$ is an \nameref{ref:equivalence-relation} on $A$, it is + \nameref{ref:symmetric} and \nameref{ref:transitive}. + Then $zRy$ and $xRy$. + By \nameref{sub:lemma-3n}, $xRy$ if and only if $[x]_R = [y]_R$, + contradicting the distinctness of $[x]_R$ and $[y]_R$. + Thus it follows $[x]_R \cap [y]_R] = \emptyset$. + + \paragraph{(ii)}% + + Let $x \in A$. + Since $R$ is an \nameref{ref:equivalence-relation} on $A$, it follows + $xRx$. + Thus $x$ is a member of some set in $\Pi$, namely $[x]_R$. + +\end{proof} + +\subsection{\sorry{Theorem 3Q}}% +\label{sub:theorem-3q} + +\begin{theorem}[3Q] + + Assume that $R$ is an equivalence relation on $A$ and that + $F \colon A \rightarrow A$. + If $F$ is compatible with $R$, then there exists a unique + $\hat{F} \colon A / R \rightarrow A / R$ such that + $$\hat{F}([x]_R) = [F(x)]_R \quad\text{for all } x \text{ in } A.$$ + If $F$ is not compatible with $R$, then no such $\hat{F}$ exists. + Analogous results apply to functions from $A \times A$ into $A$. + +\end{theorem} + \begin{proof} TODO @@ -4416,4 +4467,164 @@ Show that from the first form of the axiom of choice we can prove the second \end{proof} +\subsection{\sorry{Exercise 3.32}}% +\label{sub:exercise-3.32} + +\begin{enumerate}[(a)] + \item Show that $R$ is symmetric iff $R^{-1} \subseteq R$. + \item Show that $R$ is transitive iff $R \circ R \subseteq R$. +\end{enumerate} + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.33}}% +\label{sub:exercise-3.33} + +Show that $R$ is a symmetric and transitive relation iff $R = R^{-1} \circ R$. + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.34}}% +\label{sub:exercise-3.34} + +Assume that $\mathscr{A}$ is a nonempty set, every member of which is a + transitive relation. + +\begin{enumerate}[(a)] + \item Is the set $\bigcap{\mathscr{A}}$ a transitive relation? + \item Is $\bigcup{\mathscr{A}}$ a transitive relation? +\end{enumerate} + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.35}}% +\label{sub:exercise-3.35} + +Show that for any $R$ and $x$, we have $[x]_R = \img{R}{\{x\}}$. + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.36}}% +\label{sub:exercise-3.36} + +Assume that $f \colon A \rightarrow B$ and that $R$ is an equivalence relation + on $B$. +Define $Q$ to be the set + $$\{\left< x, y \right> \in A \times A \mid + \left< f(x), f(x) \right> \in R\}.$$ +Show that $Q$ is an equivalence relation on $A$. + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.37}}% +\label{sub:exercise-3.37} + +Assume that $\Pi$ is a partition of a set $A$. +Define the relation $R_\Pi$ as follows: + $$xR_{\Pi}y \iff (\exists B \in \Pi)(x \in B \land y \in B).$$ +Show that $R_\Pi$ is an equivalence relation on $A$. +(This is a formalized version of the discussion at the beginning of this + section.) + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.38}}% +\label{sub:exercise-3.38} + +Theorem 3P shows that $A / R$ is a partition of $A$ whenever $R$ is an + equivalence relation on $A$. +Show that if we start with the equivalence relation $R_\Pi$ of the preceding + exercise, then the partition $A / R_\Pi$ is just $\Pi$. + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.39}}% +\label{sub:exercise-3.39} + +Assume that we start with an equivalence relation $R$ on $A$ and define $\Pi$ to + be the partition $A / R$. +Show that $R_\Pi$, as defined in Exercise 37, is just $R$. + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.40}}% +\label{sub:exercise-3.40} + +Define an equivalence relation $R$ on the set $P$ of positive integers by + $$mRn \iff m \text{ and } n \text{ have the same number of prime factors}.$$ +Is there a function $f \colon P / R \rightarrow P / R$ such that + $f([n]_R) = [3n]_R$ for each $n$? + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.41}}% +\label{sub:exercise-3.41} + +Let $\mathbb{R}$ be the set of real numbers and define the relation $Q$ on + $\mathbb{R} \times \mathbb{R}$ by $\left< u, v \right>Q \left< x, y \right>$ + iff $u + y = x + v$. + +\begin{enumerate}[(a)] + \item Show that $Q$ is an equivalence relation on + $\mathbb{R} \times \mathbb{R}$. + \item Is there a function $G \colon (\mathbb{R} \times \mathbb{R}) / Q + \rightarrow (\mathbb{R} \times \mathbb{R}) / Q$ satisfying the equation + $$G([\left< x, y \right>]_Q) = [\left< x + 2y, y + 2x \right>]_Q?$$ + +\end{enumerate} + +\begin{proof} + + TODO + +\end{proof} + +\subsection{\sorry{Exercise 3.42}}% +\label{sub:exercise-3.42} + +State precisely the "analogous results" mentioned in Theorem 3Q. +(This will require extending the concept of compatibility in a suitable way.) + +\begin{proof} + + TODO + +\end{proof} + \end{document} diff --git a/Bookshelf/Enderton/Set/Chapter_3.lean b/Bookshelf/Enderton/Set/Chapter_3.lean index f972e13..8495a7f 100644 --- a/Bookshelf/Enderton/Set/Chapter_3.lean +++ b/Bookshelf/Enderton/Set/Chapter_3.lean @@ -518,7 +518,8 @@ theorem theorem_3j_a {F : Set.Relation α} {A B : Set α} intro y hy have ⟨x, hx⟩ := ran_exists hy sorry - · sorry + · intro h + sorry /-- #### Theorem 3J (b)