Add start of Enderton's "Elements of Set Theory."

finite-set-exercises
Joshua Potter 2023-05-18 14:04:20 -06:00
parent 5233126f05
commit ed981c3892
10 changed files with 220 additions and 14 deletions

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import Bookshelf.Apostol
import Bookshelf.Avigad
import Bookshelf.Enderton_Logic
import Bookshelf.Enderton.Logic
import Bookshelf.Enderton.Set
import Bookshelf.Fraleigh

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import Bookshelf.Enderton.Logic.Chapter_0

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\documentclass{report}
\input{../preamble}
\makeleancommands
\input{../../preamble}
\makeleancommands{../..}
\begin{document}

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/-! # Enderton.Logic.Chapter_0
Useful Facts About Sets
-/
namespace Enderton.Logic.Chapter_0
-- TODO
end Enderton.Logic.Chapter_0

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import Bookshelf.Enderton.Set.Chapter_1

190
Bookshelf/Enderton/Set.tex Normal file
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\documentclass{report}
\input{../../preamble}
\makeleancommands{../..}
\begin{document}
\header{Elements of Set Theory}{Herbert B. Enderton}
\tableofcontents
\begingroup
\renewcommand\thechapter{R}
\setcounter{chapter}{0}
\addtocounter{chapter}{-1}
\chapter{Reference}%
\label{chap:reference}
\section{\defined{Powerset}}%
\label{ref:powerset}
The \textbf{powerset} of some set $A$ is the set of all subsets of $A$.
\begin{definition}
\lean{Mathlib/Init/Set}{Set.powerset}
\end{definition}
\section{\defined{Principle of Extensionality}}%
\label{ref:principle-extensionality}
If $A$ and $B$ are sets such that for every object $t$,
$$t \in A \quad\text{iff}\quad t \in B,$$
then $A = B$.
\begin{axiom}
\lean{Mathlib/Init/Set}{Set.ext}
\end{axiom}
\endgroup
\chapter{Introduction}%
\label{chap:introduction}
\section{Baby Set Theory}%
\label{sec:baby-set-theory}
\subsection{\partial{Exercise 1}}%
\label{sub:baby-set-theory-1}
Which of the following become true when "$\in$" is inserted in place of the
blank?
Which become true when "$\subseteq$" is inserted?
\subsubsection{\partial{Exercise 1a}}%
\label{ssub:baby-set-theory-1a}
$\{\emptyset\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$.
\begin{proof}
Because the \textit{object} $\{\emptyset\}$ is a member of the right-hand set,
the statement is \textbf{true} in the case of "$\in$".
Because the \textit{members} of $\{\emptyset\}$ are all members of the
right-hand set, the statement is also \textbf{true} in the case of
"$\subseteq$".
\end{proof}
\subsubsection{\partial{Exercise 1b}}%
\label{ssub:baby-set-theory-1b}
$\{\emptyset\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$.
\begin{proof}
Because the \textit{object} $\{\emptyset\}$ is not a member of the right-hand
set, the statement is \textbf{false} in the case of "$\in$".
Because the \textit{members} of $\{\emptyset\}$ are all members of the
right-hand set, the statement is \textbf{true} in the case of "$\subseteq$".
\end{proof}
\subsubsection{\partial{Exercise 1c}}%
\label{ssub:baby-set-theory-1c}
$\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$.
\begin{proof}
Because the \textit{object} $\{\{\emptyset\}\}$ is not a member of the
right-hand set, the statement is \textbf{false} in the case of "$\in$".
Because the \textit{members} of $\{\{\emptyset\}\}$ are all members of the
right-hand set, the statement is \textbf{true} in the case of "$\subseteq$".
\end{proof}
\subsubsection{\partial{Exercise 1d}}%
\label{ssub:baby-set-theory-1d}
$\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$.
\begin{proof}
Because the \textit{object} $\{\{\emptyset\}\}$ is a member of the right-hand
set, the statement is \textbf{true} in the case of "$\in$".
Because the \textit{members} of $\{\{\emptyset\}\}$ are not all members of the
right-hand set, the statement is \textbf{false} in the case of
"$\subseteq$".
\end{proof}
\subsubsection{\partial{Exercise 1e}}%
\label{ssub:baby-set-theory-1e}
$\{\{\emptyset\}\} \_\_ \{\emptyset, \{\emptyset, \{\emptyset\}\}\}$.
\begin{proof}
Because the \textit{object} $\{\{\emptyset\}\}$ is not a member of the
right-hand set, the statement is \textbf{false} in the case of "$\in$".
Because the \textit{members} of $\{\{\emptyset\}\}$ are not all members of the
right-hand set, the statement is \textbf{false} in the case of
"$\subseteq$".
\end{proof}
\subsection{\partial{Exercise 2}}%
\label{sub:baby-set-theory-2}
Show that no two of the three sets $\emptyset$, $\{\emptyset\}$, and
$\{\{\emptyset\}\}$ are equal to each other.
\begin{proof}
By the \nameref{ref:principle-extensionality}, $\emptyset$ is only equal to
$\emptyset$.
This immediately shows it is not equal to the other two.
Now consider object $\emptyset$.
This object is a member of $\{\emptyset\}$ but is not a member of
$\{\{\emptyset\}\}$.
Again, by the \nameref{ref:principle-extensionality}, these two sets must be
different.
\end{proof}
\subsection{\partial{Exercise 3}}%
\label{sub:baby-set-theory-3}
Show that if $B \subseteq C$, then $\mathscr{P} B \subseteq \mathscr{P} C$.
\begin{proof}
Let $x \in \mathscr{P} B$.
By definition of the \nameref{ref:powerset}, $x$ is a subset of $B$.
By hypothesis, $B \subseteq C$.
Then $x \subseteq C$.
Again by definition of the \nameref{ref:powerset}, it follows
$x \in \mathscr{P} C$.
\end{proof}
\subsection{\partial{Exercise 4}}%
\label{sub:baby-set-theory-4}
Assume that $x$ and $y$ are members of a set $B$.
Show that $\{\{x\}, \{x, y\}\} \in \mathscr{P}\mathscr{P} B.$
\begin{proof}
Let $x$ and $y$ be members of set $B$.
Then $\{x\}$ and $\{x, y\}$ are subsets of $B$.
By definition of the \nameref{ref:powerset}, $\{x\}$ and $\{x, y\}$ are
members of $\mathscr{P} B$.
Then $\{\{x\}, \{x, y\}\}$ is a subset of $\mathscr{P} B$.
By definition of the \nameref{ref:powerset}, $\{\{x\}, \{x, y\}\}$ is a member
of $\mathscr{P}\mathscr{P} B$.
\end{proof}
\end{document}

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import Mathlib.Init.Set
/-! # Enderton.Chapter_1
Introduction
-/
namespace Enderton.Set.Chapter_1
end Enderton.Set.Chapter_1

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import Bookshelf.Enderton_Logic.Chapter_0

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/-! # Enderton.Chapter_0
Useful Facts About Sets
-/
namespace Enderton.Chapter_0
-- TODO
end Enderton.Chapter_0

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- [ ] Axler, Sheldon. Linear Algebra Done Right. Undergraduate Texts in Mathematics. Cham: Springer International Publishing, 2015.
- [ ] Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms. 3rd ed. Cambridge, Mass: MIT Press, 2009.
- [ ] Enderton, Herbert B. A Mathematical Introduction to Logic. 2nd ed. San Diego: Harcourt/Academic Press, 2001.
- [ ] Enderton, Herbert B. Elements of Set Theory. New York: Academic Press, 1977.
- [ ] Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
- [ ] Gustedt, Jens. Modern C. Shelter Island, NY: Manning Publications Co, 2020.
- [ ] Ross, Sheldon. A First Course in Probability Theory. 8th ed. Pearson Prentice Hall, n.d.