Enderton (set). Empty prompts around first few chapter 6 theorems.

finite-set-exercises
Joshua Potter 2023-08-13 09:03:51 -06:00
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@ -117,6 +117,14 @@
\lean{Mathlib/Init/Set}{Set.emptyCollection} \lean{Mathlib/Init/Set}{Set.emptyCollection}
\section{\defined{Equinumerous}}%
\hyperlabel{ref:equinumerous}
A set $A$ is \textbf{equinumerous} to a set $B$ (written $A \approx B$) if and
only if there is a one-to-one \nameref{ref:function} from $A$ onto $B$.
\lean*{Mathlib/Init/Function}{Function.Bijective}
\section{\defined{Equivalence Class}}% \section{\defined{Equivalence Class}}%
\hyperlabel{ref:equivalence-class} \hyperlabel{ref:equivalence-class}
@ -169,6 +177,14 @@
\lean{Bookshelf/Enderton/Set/Relation}{Set.Relation.fld} \lean{Bookshelf/Enderton/Set/Relation}{Set.Relation.fld}
\section{\defined{Finite Set}}%
\hyperlabel{ref:finite-set}
A set is \textbf{finite} if and only if it is \nameref{ref:equinumerous} to a
\nameref{ref:natural-number}.
\lean{Mathlib/Data/Finset/Basic}{Finset}
\section{\defined{Function}}% \section{\defined{Function}}%
\hyperlabel{ref:function} \hyperlabel{ref:function}
@ -224,6 +240,12 @@
respectively. respectively.
\end{note} \end{note}
\section{\defined{Infinite Set}}%
\hyperlabel{ref:infinite-set}
A set is \textbf{infinite} if and only if it is not a
\nameref{ref:finite-set}.
\section{\defined{Infinity Axiom}}% \section{\defined{Infinity Axiom}}%
\hyperlabel{ref:infinity-axiom} \hyperlabel{ref:infinity-axiom}
@ -8305,4 +8327,210 @@
TODO TODO
\end{proof} \end{proof}
\setcounter{chapter}{5}
\chapter{Cardinal Numbers and the Axiom of Choice}%
\hyperlabel{chap:cardinal-numbers-axiom-choice}
\section{Equinumerosity}%
\hyperlabel{sec:equinumerosity}
\subsection{\sorry{Theorem 6A}}%
\hyperlabel{sub:theorem-6a}
\begin{theorem}[6A]
For any sets $A$, $B$, and $C$,
\begin{enumerate}[(a)]
\item $A \approx A$.
\item If $A \approx B$, then $B \approx A$.
\item If $A \approx B$ and $B \approx C$, then $A \approx C$.
\end{enumerate}
\end{theorem}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Theorem 6B}}%
\hyperlabel{sub:theorem-6b}
\begin{theorem}[6B]
No set is equinumerous to its powerset.
\end{theorem}
\begin{proof}
TODO
\end{proof}
\section{Finite Sets}%
\hyperlabel{sec:finite-sets}
\subsection{\sorry{Pigeonhole Principle}}%
\hyperlabel{sub:pigeonhole-principle}
\begin{theorem}
No natural number is equinumerous to a proper subset of itself.
\end{theorem}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Corollary 6C}}%
\hyperlabel{sub:corollary-6c}
\begin{corollary}[6C]
No finite set is equinumerous to a proper subset of itself.
\end{corollary}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Corollary 6D}}%
\hyperlabel{sub:corollary-6d}
\begin{corollary}[6D]
\begin{enumerate}[(a)]
\item Any set equinumerous to a proper subset of itself is infinite.
\item The set $\omega$ is infinite.
\end{enumerate}
\end{corollary}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Corollary 6E}}%
\hyperlabel{sub:corollary-6e}
\begin{corollary}[6E]
Any finite set is equinumerous to a unique natural number.
\end{corollary}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Lemma 6F}}%
\hyperlabel{sub:lemma-6f}
\begin{lemma}[6F]
If $C$ is a proper subset of a natural number $n$, then $C \approx m$ for
some $m$ less than $n$.
\end{lemma}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Corollary 6G}}%
\hyperlabel{sub:corollary-6g}
\begin{corollary}[6G]
Any subset of a finite set is finite.
\end{corollary}
\begin{proof}
TODO
\end{proof}
\section{Exercises 6}%
\hyperlabel{sec:exercises-6}
\subsection{\sorry{Exercise 6.1}}%
\hyperlabel{sub:exercise-6-1}
Show that the equation $$f(m, n) = 2^m(2n + 1) - 1$$ defines a one-one-one
correspondence between $\omega \times \omega$ and $\omega$.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.2}}%
\hyperlabel{sub:exercise-6-2}
Show that in Fig. 32 we have:
\begin{align*}
J(m, n)
& = [1 + 2 + \cdots + (m + n)] + m \\
& = \frac{1}{2}[(m + n)^2 + 3m + n].
\end{align*}
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.3}}%
\hyperlabel{sub:exercise-6-3}
Find a one-to-one correspondence between the open unit interval $\ioo{0}{1}$
and $\mathbb{R}$ that takes rationals to rationals and irrationals to
irrationals.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.4}}%
\hyperlabel{sub:exercise-6-4}
Construct a one-to-one correspondence between the closed unit interval
$$\icc{0}{1} = \{x \in \mathbb{R} \mid 0 \leq x \leq 1\}$$
and the open unit interval $\ioo{0}{1}$.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.5}}%
\hyperlabel{sub:exercise-6-5}
Prove \nameref{sub:theorem-6a}.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.6}}%
\hyperlabel{sub:exercise-6-6}
Let $\kappa$ be a nonzero cardinal number.
Show there does not exist a set to which every set of cardinality $\kappa$
belongs.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.7}}%
\hyperlabel{sub:exercise-6-7}
Assume that $A$ is finite and $f \colon A \rightarrow A$.
Show that $f$ is one-to-one iff $\ran{f} = A$.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.8}}%
\hyperlabel{sub:exercise-6-8}
Prove that the union of two finite sets is finite, without any use of
arithmetic.
\begin{proof}
TODO
\end{proof}
\subsection{\sorry{Exercise 6.9}}%
\hyperlabel{sub:exercise-6-9}
Prove that the Cartesian product of two finite sets is finite, without any use
of arithmetic.
\begin{proof}
TODO
\end{proof}
\end{document} \end{document}