Enderton. Add equivalence relation problem prompts.

finite-set-exercises
Joshua Potter 2023-07-02 12:26:09 -06:00
parent 48ed3cd59e
commit 140195a8ee
2 changed files with 214 additions and 2 deletions

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@ -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 any set $I$ and any function $H$ with domain $I$, if $H(i) \neq \emptyset$
for all $i \in I$, then $$\bigtimes_{i \in I} H(i) \neq \emptyset.$$ 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}}% \section{\defined{Composition}}%
\label{ref:composition} \label{ref:composition}
@ -3269,7 +3276,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
\end{proof} \end{proof}
\subsection{\sorry{Theorem 3P}}% \subsection{\pending{Theorem 3P}}%
\label{sub:theorem-3p} \label{sub:theorem-3p}
\begin{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} \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} \begin{proof}
TODO TODO
@ -4416,4 +4467,164 @@ Show that from the first form of the axiom of choice we can prove the second
\end{proof} \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} \end{document}

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@ -518,7 +518,8 @@ theorem theorem_3j_a {F : Set.Relation α} {A B : Set α}
intro y hy intro y hy
have ⟨x, hx⟩ := ran_exists hy have ⟨x, hx⟩ := ran_exists hy
sorry sorry
· sorry · intro h
sorry
/-- #### Theorem 3J (b) /-- #### Theorem 3J (b)