2614 lines
80 KiB
TeX
2614 lines
80 KiB
TeX
\documentclass{report}
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\usepackage{graphicx}
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\graphicspath{{./Set/images/}}
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\input{../../preamble}
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\makeleancommands{../..}
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\begin{document}
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\header{Elements of Set Theory}{Herbert B. Enderton}
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\tableofcontents
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\begingroup
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\renewcommand\thechapter{R}
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\setcounter{chapter}{0}
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\addtocounter{chapter}{-1}
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\chapter{Reference}%
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\label{chap:reference}
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\section{\defined{Empty Set Axiom}}%
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\label{ref:empty-set-axiom}
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There is a set having no members:
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$$\exists B, \forall x, x \not\in B.$$
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\begin{axiom}
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\lean*{Mathlib/Init/Set}{Set.emptyCollection}
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\end{axiom}
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\section{\defined{Extensionality Axiom}}%
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\label{ref:extensionality-axiom}
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If two sets have exactly the same members, then they are equal:
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$$\forall A, \forall B,
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\left[\forall x, (x \in A \iff x \in B) \Rightarrow A = B\right].$$
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\begin{axiom}
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\lean*{Mathlib/Init/Set}{Set.ext}
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\end{axiom}
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\section{\defined{Ordered Pair}}%
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\label{ref:ordered-pair}
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For any sets $u$ and $v$, the \textbf{ordered pair} $\left< u, v \right>$ is
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the set $\{\{u\}, \{u, v\}\}$.
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\begin{definition}
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\lean*{Common/Set/OrderedPair}{OrderedPair}
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\end{definition}
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\section{\defined{Pair Set}}%
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\label{ref:pair-set}
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For any sets $u$ and $v$, the \textbf{pair set $\{u, v\}$} is the set whose
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only members are $u$ and $v$.
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\begin{definition}
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\statementpadding
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\lean*{Mathlib/Init/Set}{Set.insert}
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\lean*{Mathlib/Init/Set}{Set.singleton}
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\end{definition}
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\section{\defined{Pairing Axiom}}%
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\label{ref:pairing-axiom}
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For any sets $u$ and $v$, there is a set having as members just $u$ and $v$:
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$$\forall u, \forall v, \exists B, \forall x,
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(x \in B \iff x = u \text{ or } x = v).$$
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\begin{axiom}
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\statementpadding
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\lean*{Mathlib/Init/Set}{Set.insert}
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\lean*{Mathlib/Init/Set}{Set.singleton}
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\end{axiom}
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\section{\defined{Power Set}}%
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\label{ref:power-set}
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For any set $a$, the \textbf{power set $\powerset{a}$} is the set whose members
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are exactly the subsets of $a$.
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\begin{definition}
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\lean*{Mathlib/Init/Set}{Set.powerset}
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\end{definition}
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\section{\defined{Power Set Axiom}}%
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\label{ref:power-set-axiom}
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For any set $a$, there is a set whose members are exactly the subsets of $a$:
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$$\forall a, \exists B, \forall x, (x \in B \iff x \subseteq a).$$
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\begin{axiom}
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\lean*{Mathlib/Init/Set}{Set.powerset}
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\end{axiom}
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\section{\defined{Subset Axioms}}%
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\label{ref:subset-axioms}
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For each formula $\phi$ not containing $B$, the following is an axiom:
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$$\forall t_1, \cdots \forall t_k, \forall c,
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\exists B, \forall x, (x \in B \iff x \in c \land \phi).$$
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\begin{axiom}
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\lean*{Mathlib/Init/Set}{Set.Subset}
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\end{axiom}
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\section{\defined{Symmetric Difference}}%
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\label{ref:symmetric-difference}
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The \textbf{symmetric difference} $A + B$ of sets $A$ and $B$ is the set
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$(A - B) \cup (B - A)$.
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\begin{definition}
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\lean*{Mathlib/Data/Set/Basic}{symmDiff\_def}
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\end{definition}
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\section{\defined{Union Axiom}}%
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\label{ref:union-axiom}
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For any set $A$, there exists a set $B$ whose elements are exactly the members
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of the members of $A$:
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$$\forall A, \exists B, \forall x \left[ x \in B \iff (\exists b \in A) x \in b \right]$$
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\begin{axiom}
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\lean*{Mathlib/Data/Set/Lattice}{Set.sUnion}
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\end{axiom}
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\section{\defined{Union Axiom, Preliminary Form}}%
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\label{ref:union-axiom-preliminary-form}
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For any sets $a$ and $b$, there is a set whose members are those sets belonging
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either to $a$ or to $b$ (or both):
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$$\forall a, \forall b, \exists B, \forall x,
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(x \in B \iff x \in a \text{ or } x \in b).$$
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\begin{axiom}
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\lean*{Mathlib/Init/Set}{Set.union}
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\end{axiom}
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\endgroup
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\chapter{Introduction}%
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\label{chap:introduction}
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\section{Baby Set Theory}%
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\label{sec:baby-set-theory}
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\subsection{\verified{Exercise 1.1}}%
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\label{sub:exercise-1.1}
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Which of the following become true when "$\in$" is inserted in place of the
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blank?
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Which become true when "$\subseteq$" is inserted?
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\subsubsection{\verified{Exercise 1.1a}}%
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\label{ssub:exercise-1.1a}
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$\{\emptyset\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_1a}
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Because the \textit{object} $\{\emptyset\}$ is a member of the right-hand set,
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the statement is \textbf{true} in the case of "$\in$".
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Because the \textit{members} of $\{\emptyset\}$ are all members of the
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right-hand set, the statement is also \textbf{true} in the case of
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"$\subseteq$".
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\end{proof}
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\subsubsection{\verified{Exercise 1.1b}}%
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\label{ssub:exercise-1.11b}
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$\{\emptyset\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_1b}
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Because the \textit{object} $\{\emptyset\}$ is not a member of the right-hand
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set, the statement is \textbf{false} in the case of "$\in$".
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Because the \textit{members} of $\{\emptyset\}$ are all members of the
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right-hand set, the statement is \textbf{true} in the case of "$\subseteq$".
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\end{proof}
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\subsubsection{\verified{Exercise 1.1c}}%
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\label{ssub:exercise-1.1c}
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$\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\emptyset\}\}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_1c}
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Because the \textit{object} $\{\{\emptyset\}\}$ is not a member of the
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right-hand set, the statement is \textbf{false} in the case of "$\in$".
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Because the \textit{members} of $\{\{\emptyset\}\}$ are all members of the
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right-hand set, the statement is \textbf{true} in the case of "$\subseteq$".
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\end{proof}
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\subsubsection{\verified{Exercise 1.1d}}%
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\label{ssub:exercise-1.1d}
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$\{\{\emptyset\}\} \_\_\_\_ \{\emptyset, \{\{\emptyset\}\}\}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_1d}
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Because the \textit{object} $\{\{\emptyset\}\}$ is a member of the right-hand
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set, the statement is \textbf{true} in the case of "$\in$".
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Because the \textit{members} of $\{\{\emptyset\}\}$ are not all members of the
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right-hand set, the statement is \textbf{false} in the case of
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"$\subseteq$".
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\end{proof}
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\subsubsection{\verified{Exercise 1.1e}}%
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\label{ssub:exercise-1.1e}
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$\{\{\emptyset\}\} \_\_ \{\emptyset, \{\emptyset, \{\emptyset\}\}\}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_1e}
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Because the \textit{object} $\{\{\emptyset\}\}$ is not a member of the
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right-hand set, the statement is \textbf{false} in the case of "$\in$".
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Because the \textit{members} of $\{\{\emptyset\}\}$ are not all members of the
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right-hand set, the statement is \textbf{false} in the case of
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"$\subseteq$".
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\end{proof}
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\subsection{\verified{Exercise 1.2}}%
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\label{sub:exercise-1.2}
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Show that no two of the three sets $\emptyset$, $\{\emptyset\}$, and
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$\{\{\emptyset\}\}$ are equal to each other.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_2}
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By the \nameref{ref:extensionality-axiom}, $\emptyset$ is only equal to
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$\emptyset$.
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This immediately shows it is not equal to the other two.
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Now consider object $\emptyset$.
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This object is a member of $\{\emptyset\}$ but is not a member of
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$\{\{\emptyset\}\}$.
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Again, by the \nameref{ref:extensionality-axiom}, these two sets must be
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different.
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\end{proof}
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\subsection{\verified{Exercise 1.3}}%
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\label{sub:exercise-1.3}
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Show that if $B \subseteq C$, then $\powerset{B} \subseteq \powerset{C}$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_3}
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Let $x \in \powerset{B}$.
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By definition of the \nameref{ref:power-set}, $x$ is a subset of $B$.
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By hypothesis, $B \subseteq C$.
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Then $x \subseteq C$.
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Again by definition of the \nameref{ref:power-set}, it follows
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$x \in \powerset{C}$.
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\end{proof}
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\subsection{\verified{Exercise 1.4}}%
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\label{sub:exercise-1.4}
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Assume that $x$ and $y$ are members of a set $B$.
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Show that $\{\{x\}, \{x, y\}\} \in \powerset{\powerset{B}}.$
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_1}
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{Enderton.Set.Chapter\_1.exercise\_1\_4}
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Let $x$ and $y$ be members of set $B$.
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Then $\{x\}$ and $\{x, y\}$ are subsets of $B$.
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By definition of the \nameref{ref:power-set}, $\{x\}$ and $\{x, y\}$ are
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members of $\powerset{B}$.
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Then $\{\{x\}, \{x, y\}\}$ is a subset of $\powerset{B}$.
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By definition of the \nameref{ref:power-set}, $\{\{x\}, \{x, y\}\}$ is a
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member of $\powerset{\powerset{B}}$.
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\end{proof}
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\section{Sets - An Informal View}%
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\label{sec:sets-informal-view}
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\subsection{\partial{Exercise 2.1}}%
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\label{sub:exercise-2.1}
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Define the rank of a set $c$ to be the least $\alpha$ such that
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$c \subseteq V_\alpha$.
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Compute the rank of $\{\{\emptyset\}\}$.
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Compute the rank of
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$\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$.
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\begin{proof}
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We first compute the values of $V_n$ for $0 \leq n \leq 3$ under the
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assumption the set of atoms $A$ at the bottom of the hierarchy is empty.
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\begin{align*}
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V_0 & = \emptyset \\
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V_1 & = V_0 \cup \powerset{V_0} \\
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& = \emptyset \cup \{\emptyset\} \\
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& = \{\emptyset\} \\
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V_2 & = V_1 \cup \powerset{V_1} \\
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& = \{\emptyset\} \cup \powerset{\{\emptyset\}} \\
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& = \{\emptyset\} \cup \{\emptyset, \{\emptyset\}\} \\
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& = \{\emptyset, \{\emptyset\}\} \\
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V_3 & = V_2 \cup \powerset{V_2} \\
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& = \{\emptyset, \{\emptyset\}\} \cup
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\powerset{\{\emptyset, \{\emptyset\}\}} \\
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& = \{\emptyset, \{\emptyset\}\} \cup
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\{\emptyset,
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\{\emptyset\},
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\{\{\emptyset\}\},
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\{\emptyset, \{\emptyset\}\}\} \\
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& = \{\emptyset,
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\{\emptyset\},
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\{\{\emptyset\}\},
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\{\emptyset, \{\emptyset\}\}\}
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\end{align*}
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It then immediately follows $\{\{\emptyset\}\}$ has rank $2$ and
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$\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$ has rank $3$.
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\end{proof}
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\subsection{\partial{Exercise 2.2}}%
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\label{sub:exercise-2.2}
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We have stated that $V_{\alpha + 1} = A \cup \powerset{V_\alpha}$.
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Prove this at least for $\alpha < 3$.
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\begin{proof}
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Let $A$ be the set of atoms in our set hierarchy.
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Let $P(n)$ be the predicate, "$V_{n + 1} = A \cup \powerset{V_n}$."
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We prove $P(n)$ holds true for all natural numbers $n \geq 1$ via induction.
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\paragraph{Base Case}%
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Let $n = 1$.
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By definition, $V_1 = V_0 \cup \powerset{V_0}$.
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By definition, $V_0 = A$.
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Therefore $V_1 = A \cup \powerset{V_0}$.
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This proves $P(1)$ holds true.
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\paragraph{Induction Step}%
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Suppose $P(n)$ holds true for some $n \geq 1$.
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Consider $V_{n+1}$.
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By definition, $V_{n+1} = V_n \cup \powerset{V_n}$.
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Therefore, by the induction hypothesis,
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\begin{align}
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V_{n+1}
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& = V_n \cup \powerset{V_n}
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\nonumber \\
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& = (A \cup \powerset{V_{n-1}}) \cup \powerset{V_n}
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\nonumber \\
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& = A \cup (\powerset{V_{n-1}} \cup \powerset{V_n})
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\label{sub:exercise-2.2-eq1}
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\end{align}
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But $V_{n-1}$ is a subset of $V_n$.
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\nameref{sub:exercise-1.3} then implies
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$\powerset{V_{n-1}} \subseteq \powerset{V_n}$.
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This means \eqref{sub:exercise-2.2-eq1} can be simplified to
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$$V_{n+1} = A \cup \powerset{V_n},$$
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proving $P(n+1)$ holds true.
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\paragraph{Conclusion}%
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By mathematical induction, it follows for all $n \geq 1$, $P(n)$ is true.
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\end{proof}
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\subsection{\partial{Exercise 2.3}}%
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\label{sub:exercise-2.3}
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List all the members of $V_3$.
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List all the members of $V_4$.
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(It is to be assumed here that there are no atoms.)
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\begin{proof}
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As seen in the proof of \nameref{sub:exercise-2.1},
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$$V_3 = \{
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\emptyset,
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\{\emptyset\},
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\{\{\emptyset\}\},
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\{\emptyset, \{\emptyset\}\}
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\}.$$
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By \nameref{sub:exercise-2.2}, $V_4 = \powerset{V_3}$ (since it is assumed
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there are no atoms).
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Thus
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\begin{align*}
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& V_4 = \{ \\
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& \qquad \emptyset, \\
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& \qquad \{\emptyset\}, \\
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& \qquad \{\{\emptyset\}\}, \\
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& \qquad \{\{\{\emptyset\}\}\}, \\
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& \qquad \{\{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\emptyset\}\}, \\
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& \qquad \{\emptyset, \{\{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\{\emptyset\}, \{\{\emptyset\}\}\}, \\
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& \qquad \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \\
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& \qquad \{\{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\}, \\
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& \qquad \{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset, \{\emptyset\}\}\} \\
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& \}.
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\end{align*}
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\end{proof}
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\chapter{Axioms and Operations}%
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\label{chap:axioms-operations}
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\section{Axioms}%
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\label{sec:axioms}
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\subsection{\partial{Theorem 2A}}%
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\label{sub:theorem-2a}
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\begin{theorem}[2A]
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There is no set to which every set belongs.
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\note{This was revisited after reading Enderton's proof prior.}
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\end{theorem}
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\begin{proof}
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Let $A$ be an arbitrary set.
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Define $B = \{ x \in A \mid x \not\in x \}$.
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By the \nameref{ref:subset-axioms}, $B$ is a set.
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Then $$B \in B \iff B \in A \land B \not\in B.$$
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If $B \in A$, then $B \in B \iff B \not\in B$, a contradiction.
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Thus $B \not\in A$.
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Since this process holds for any set $A$, there must exist no set to which
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every set belongs.
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\end{proof}
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\subsection{\partial{Theorem 2B}}%
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\label{sub:theorem-2b}
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\begin{theorem}[2B]
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For any nonempty set $A$, there exists a unique set $B$ such that for any
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$x$, $$x \in B \iff x \text{ belongs to every member of } A.$$
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\end{theorem}
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\begin{proof}
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Suppose $A$ is a nonempty set.
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This ensures the statement we are trying to prove does not vacuously hold for
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all sets $x$ (which would yield a contradiction due to
|
|
\nameref{sub:theorem-2b}).
|
|
By the \nameref{ref:union-axiom}, $\bigcup A$ is a set.
|
|
Define $$B = \{ x \in \bigcup A \mid (\forall b \in A), x \in b \}.$$
|
|
By the \nameref{ref:subset-axioms}, $B$ is indeed a set.
|
|
By construction,
|
|
$$\forall x, x \in B \iff x \text{ belongs to every member of } A.$$
|
|
By the \nameref{ref:extensionality-axiom}, $B$ is unique.
|
|
|
|
\end{proof}
|
|
|
|
\section{Exercises 3}%
|
|
\label{sec:exercises-3}
|
|
|
|
\subsection{\verified{Exercise 3.1}}%
|
|
\label{sub:exercise-3.1}
|
|
|
|
Assume that $A$ is the set of integers divisible by $4$.
|
|
Similarly assume that $B$ and $C$ are the sets of integers divisible by $9$ and
|
|
$10$, respectively.
|
|
What is in $A \cap B \cap C$?
|
|
|
|
\begin{answer}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_1}
|
|
|
|
The set of integers divisible by $4$, $9$, and $10$.
|
|
|
|
\end{answer}
|
|
|
|
\subsection{\verified{Exercise 3.2}}%
|
|
\label{sub:exercise-3.2}
|
|
|
|
Give an example of sets $A$ and $B$ for which $\bigcup A = \bigcup B$ but
|
|
$A \neq B$.
|
|
|
|
\begin{answer}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_2}
|
|
|
|
Let $A = \{\{1\}, \{2\}\}$ and $B = \{\{1, 2\}\}$.
|
|
|
|
\end{answer}
|
|
|
|
\subsection{\verified{Exercise 3.3}}%
|
|
\label{sub:exercise-3.3}
|
|
|
|
Show that every member of a set $A$ is a subset of $\bigcup A$.
|
|
(This was stated as an example in this section.)
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_3}
|
|
|
|
Let $x \in A$.
|
|
By definition, $$\bigcup A = \{ y \mid (\exists b \in A) y \in b\}.$$
|
|
Then $\{ y \mid y \in x\} \subseteq \bigcup A$.
|
|
But $\{ y \mid y \in x\} = x$.
|
|
Thus $x \subseteq \bigcup A$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.4}}%
|
|
\label{sub:exercise-3.4}
|
|
|
|
Show that if $A \subseteq B$, then $\bigcup A \subseteq \bigcup B$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_4}
|
|
|
|
Let $A$ and $B$ be sets such that $A \subseteq B$.
|
|
Let $x \in \bigcup A$.
|
|
By definition of the union, there exists some $b \in A$ such that $x \in b$.
|
|
By definition of the subset, $b \in B$.
|
|
This immediatley implies $x \in \bigcup B$.
|
|
Since this holds for all $x \in \bigcup A$, it follows
|
|
$\bigcup A \subseteq \bigcup B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.5}}%
|
|
\label{sub:exercise-3.5}
|
|
|
|
Assume that every member of $\mathscr{A}$ is a subset of $B$.
|
|
Show that $\bigcup \mathscr{A} \subseteq B$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_5}
|
|
|
|
Let $x \in \bigcup \mathscr{A}$.
|
|
By definition,
|
|
$$\bigcup \mathscr{A} = \{ y \mid (\exists b \in A)y \in b \}.$$
|
|
Then there exists some $b \in A$ such that $x \in b$.
|
|
By hypothesis, $b \subseteq B$.
|
|
Thus $x$ must also be a member of $B$.
|
|
Since this holds for all $x \in \bigcup \mathscr{A}$, it follows
|
|
$\bigcup \mathscr{A} \subseteq B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.6a}}%
|
|
\label{sub:exercise-3.6a}
|
|
|
|
Show that for any set $A$, $\bigcup \powerset{A} = A$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_6a}
|
|
|
|
We prove that (i) $\bigcup \powerset{A} \subseteq A$ and (ii)
|
|
$A \subseteq \bigcup \powerset{A}$.
|
|
|
|
\paragraph{(i)}%
|
|
\label{par:exercise-3.6a-i}
|
|
|
|
By definition, the \nameref{ref:power-set} of $A$ is the set of all subsets
|
|
of $A$.
|
|
In other words, every member of $\powerset{A}$ is a subset of $A$.
|
|
By \nameref{sub:exercise-3.5}, $\bigcup \powerset{A} \subseteq A$.
|
|
|
|
\paragraph{(ii)}%
|
|
\label{par:exercise-3.6a-ii}
|
|
|
|
Let $x \in A$.
|
|
By definition of the power set of $A$, $\{x\} \in \powerset{A}$.
|
|
By definition of the union,
|
|
$$\bigcup \powerset{A} =
|
|
\{ y \mid (\exists b \in \powerset{A}), y \in b).$$
|
|
Since $x \in \{x\}$ and $\{x\} \in \powerset{A}$, it follows
|
|
$x \in \bigcup \powerset{A}$.
|
|
Thus $A \subseteq \bigcup \powerset{A}$.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
By \nameref{par:exercise-3.6a-i} and \nameref{par:exercise-3.6a-ii},
|
|
$\bigcup \powerset{A} = A$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.6b}}%
|
|
\label{sub:exercise-3.6b}
|
|
|
|
Show that $A \subseteq \powerset{\bigcup A}$.
|
|
Under what conditions does equality hold?
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_6b}
|
|
|
|
Let $x \in A$.
|
|
By \nameref{sub:exercise-3.3}, $x$ is a subset of $\bigcup A$.
|
|
By the definition of the \nameref{ref:power-set},
|
|
$$\powerset{\bigcup A} = \{ y \mid y \subseteq \bigcup A \}.$$
|
|
Therefore $x \in \powerset{\bigcup A}$.
|
|
Since this holds for all $x \in A$, $A \subseteq \powerset{\bigcup A}$.
|
|
|
|
\suitdivider
|
|
|
|
We show equality holds if and only if there exists some set $B$ such that
|
|
$A = \powerset{B}$.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
\label{par:exercise-3.6b-right}
|
|
|
|
Suppose $A = \powerset{\bigcup A}$.
|
|
Then our statement immediately follows by settings $B = \bigcup A$.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
\label{par:exercise-3.6b-left}
|
|
|
|
Suppose there exists some set $B$ such that $A = \powerset{B}$.
|
|
Therefore
|
|
\begin{align*}
|
|
\powerset{\bigcup A}
|
|
& = \powerset{\left(\bigcup {\powerset {B}}\right)} \\
|
|
& = \powerset{B} & \textref{sub:exercise-3.6a} \\
|
|
& = A.
|
|
\end{align*}
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
By \nameref{par:exercise-3.6b-right} and \nameref{par:exercise-3.6b-left},
|
|
$A = \powerset{\bigcup A}$ if and only if there exists some set $B$ such
|
|
that $A = \powerset{B}$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.7a}}%
|
|
\label{sub:exercise-3.7a}
|
|
|
|
Show that for any sets $A$ and $B$,
|
|
$$\powerset{A} \cap \powerset{B} = \powerset{(A \cap B)}.$$
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_7a}
|
|
|
|
Let $A$ and $B$ be arbitrary sets. We show that
|
|
$\powerset{A} \cap \powerset{B} \subseteq \powerset{(A \cap B)}$ and then
|
|
show that $\powerset{A} \cap \powerset{B} \supseteq \powerset{(A \cap B)}$.
|
|
|
|
\paragraph{($\subseteq$)}%
|
|
|
|
Let $x \in \powerset{A} \cap \powerset{B}$.
|
|
That is, $x \in \powerset{A}$ and $x \in \powerset{B}$.
|
|
By the definition of the \nameref{ref:power-set},
|
|
\begin{align*}
|
|
\powerset{A} & = \{ y \mid y \subseteq A \} \\
|
|
\powerset{B} & = \{ y \mid y \subseteq B \}
|
|
\end{align*}
|
|
Thus $x \subseteq A$ and $x \subseteq B$, meaning $x \subseteq A \cap B$.
|
|
But then $x \in \powerset{(A \cap B)}$, the set of all subsets of
|
|
$A \cap B$.
|
|
Since this holds for all $x \in \powerset{A} \cap \powerset{B}$, it follows
|
|
$$\powerset{A} \cap \powerset{B} \subseteq \powerset{(A \cap B)}.$$
|
|
|
|
\paragraph{($\supseteq$)}%
|
|
|
|
Let $x \in \powerset{(A \cap B)}$.
|
|
By the definition of the \nameref{ref:power-set},
|
|
$$\powerset{(A \cap B)} = \{ y \mid y \subseteq A \cap B \}.$$
|
|
Thus $x \subseteq A \cap B$, meaning $x \subseteq A$ and $x \subseteq B$.
|
|
But this implies $x \in \powerset{A}$, the set of all subsets of $A$.
|
|
Likewise $x \in \powerset{B}$, the set of all subsets of $B$.
|
|
Thus $x \in \powerset{A} \cap \powerset{B}$.
|
|
Since this holds for all $x \in \powerset{(A \cap B)}$, it follows
|
|
$$\powerset{(A \cap B)} \subseteq \powerset{A} \cap \powerset{B}.$$
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
Since each side of our identity is a subset of the other,
|
|
$$\powerset{(A \cap B)} = \powerset{A} \cap \powerset{B}.$$
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.7b}}%
|
|
\label{sub:exercise-3.7b}
|
|
|
|
Show that $\powerset{A} \cup \powerset{B} \subseteq \powerset{(A \cup B)}$.
|
|
Under what conditions does equality hold?
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_7b\_i}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_7b\_ii}
|
|
|
|
Let $x \in \powerset{A} \cup \powerset{B}$.
|
|
By definition, $x \in \powerset{A}$ or $x \in \powerset{B}$ (or both).
|
|
By the definition of the \nameref{ref:power-set},
|
|
\begin{align*}
|
|
\powerset{A} &= \{ y \mid y \subseteq A \} \\
|
|
\powerset{B} &= \{ y \mid y \subseteq B \}.
|
|
\end{align*}
|
|
Thus $x \subseteq A$ or $x \subseteq B$.
|
|
Therefore $x \subseteq A \cup B$.
|
|
But then $x \in \powerset{(A \cup B)}$, the set of all subsets of $A \cup B$.
|
|
|
|
\suitdivider
|
|
|
|
We show equality holds if and only if one of $A$ or $B$ is a subset of the
|
|
other.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
\label{par:exercise-3.7b-right}
|
|
|
|
Suppose
|
|
\begin{equation}
|
|
\label{sub:exercise-3.7b-eq1}
|
|
\powerset{A} \cup \powerset{B} = \powerset{(A \cup B)}.
|
|
\end{equation}
|
|
By the definition of the \nameref{ref:power-set},
|
|
$A \cup B \in \powerset{(A \cup B)}$.
|
|
Then \eqref{sub:exercise-3.7b-eq1} implies
|
|
$A \cup B \in \powerset{A} \cup \powerset{B}$.
|
|
That is, $A \cup B \in \powerset{A}$ or $A \cup B \in \powerset{B}$ (or
|
|
both).
|
|
|
|
For the sake of contradiction, suppose $A \not\subseteq B$ and
|
|
$B \not\subseteq A$.
|
|
Then there exists an element $x \in A$ such that $x \not\in B$ and there
|
|
exists an element $y \in B$ such that $y \not\in A$.
|
|
But then $A \cup B \not\in \powerset{A}$ since $y$ cannot be a member of a
|
|
member of $\powerset{A}$.
|
|
Likewise, $A \cup B \not\in \powerset{B}$ since $x$ cannot be a member of a
|
|
member of $\powerset{B}$.
|
|
Therefore our assumption is incorrect.
|
|
In other words, $A \subseteq B$ or $B \subseteq A$.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
\label{par:exercise-3.7b-left}
|
|
|
|
WLOG, suppose $A \subseteq B$.
|
|
Then, by \nameref{sub:exercise-1.3}, $\powerset{A} \subseteq \powerset{B}$.
|
|
Thus
|
|
\begin{align*}
|
|
\powerset{A} \cup \powerset{B}
|
|
& = \powerset{B} \\
|
|
& = \powerset{A \cup B}.
|
|
\end{align*}
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
By \nameref{par:exercise-3.7b-right} and \nameref{par:exercise-3.7b-left},
|
|
it follows
|
|
$\powerset{A} \cup \powerset{B} \subseteq \powerset{(A \cup B)}$ if and
|
|
only if $A \subseteq B$ or $B \subseteq A$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 3.8}}%
|
|
\label{sub:exercise-3.8}
|
|
|
|
Show that there is no set to which every singleton (that is, every set of the
|
|
form $\{x\}$) belongs.
|
|
[\textit{Suggestion}: Show that from such a set, we could construct a set to
|
|
which every set belonged.]
|
|
|
|
\begin{proof}
|
|
|
|
We proceed by contradiction.
|
|
Suppose there existed a set $A$ consisting of every singleton.
|
|
Then the \nameref{ref:union-axiom} suggests $\bigcup A$ is a set.
|
|
But this set is precisely the class of all sets, which is \textit{not} a set.
|
|
Thus our original assumption was incorrect.
|
|
That is, there is no set to which every singleton belongs.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 3.9}}%
|
|
\label{sub:exercise-3.9}
|
|
|
|
Give an example of sets $a$ and $B$ for which $a \in B$ but
|
|
$\powerset{a} \not\in \powerset{B}$.
|
|
|
|
\begin{answer}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_9}
|
|
|
|
Let $a = \{1\}$ and $B = \{\{1\}\}$.
|
|
Then
|
|
\begin{align*}
|
|
\powerset{a} & = \{\emptyset, \{1\}\} \\
|
|
\powerset{B} & = \{\emptyset, \{\{1\}\}\}.
|
|
\end{align*}
|
|
It immediately follows that $\powerset{a} \not\in \powerset{B}$.
|
|
|
|
\end{answer}
|
|
|
|
\subsection{\verified{Exercise 3.10}}%
|
|
\label{sub:exercise-3.10}
|
|
|
|
Show that if $a \in B$, then $\powerset{a} \in \powerset{\powerset{\bigcup B}}$.
|
|
[\textit{Suggestion}: If you need help, look in the Appendix.]
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_3\_10}
|
|
|
|
Suppose $a \in B$.
|
|
By \nameref{sub:exercise-3.3}, $a \subseteq \bigcup B$.
|
|
By \nameref{sub:exercise-1.3}, $\powerset{a} \subseteq \powerset{\bigcup B}$.
|
|
By the definition of the \nameref{ref:power-set},
|
|
$$\powerset{\powerset{\bigcup B}} =
|
|
\{ y \mid y \subseteq \powerset{\bigcup B} \}.$$
|
|
Therefore $\powerset{a} \in \powerset{\powerset{\bigcup B}}$.
|
|
|
|
\end{proof}
|
|
|
|
\section{Algebra of Sets}%
|
|
\label{sec:algebra-sets}
|
|
|
|
\subsection{\verified{Commutative Laws}}%
|
|
\label{sub:commutative-laws}
|
|
|
|
For any sets $A$ and $B$,
|
|
\begin{align*}
|
|
A \cup B = B \cup A \\
|
|
A \cap B = B \cap A
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.union\_comm}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.inter\_comm}
|
|
|
|
\noindent Let $A$ and $B$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \cup B = B \cup A$
|
|
\item $A \cap B = B \cap A$.
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By the definition of the union of sets,
|
|
\begin{align*}
|
|
A \cup B
|
|
& = \{ x \mid x \in A \lor x \in B \} \\
|
|
& = \{ x \mid x \in B \lor x \in A \} \\
|
|
& = B \cup A.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By the definition of the intersection of sets,
|
|
\begin{align*}
|
|
A \cap B
|
|
& = \{ x \mid x \in A \land x \in B \} \\
|
|
& = \{ x \mid x \in B \land x \in A \} \\
|
|
& = B \land A.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Associative Laws}}%
|
|
\label{sub:associative-laws}
|
|
|
|
For any sets $A$, $B$ and $C$,
|
|
\begin{align*}
|
|
A \cup (B \cup C) & = (A \cup B) \cup C \\
|
|
A \cap (B \cap C) & = (A \cap B) \cap C
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.union\_assoc}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.inter\_assoc}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \cup (B \cup C) = (A \cup B) \cup C$
|
|
\item $A \cap (B \cap C) = (A \cap B) \cap C$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By the definition of the union of sets,
|
|
\begin{align*}
|
|
A \cup (B \cup C)
|
|
& = \{ x \mid x \in A \lor x \in (B \cup C) \} \\
|
|
& = \{ x \mid x \in A \lor x \in \{ y \mid y \in B \lor C \}\} \\
|
|
& = \{ x \mid x \in A \lor (x \in B \lor x \in C) \} \\
|
|
& = \{ x \mid (x \in A \lor x \in B) \lor x \in C \} \\
|
|
& = \{ x \mid x \in \{ y \mid y \in A \lor y \in B \} \lor
|
|
x \in C \} \\
|
|
& = \{ x \mid x \in (A \cup B) \lor x \in C \} \\
|
|
& = (A \cup B) \cup C.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By the definition of the intersection of sets,
|
|
\begin{align*}
|
|
A \cap (B \cap C)
|
|
& = \{ x \mid x \in A \land x \in (B \cap C) \} \\
|
|
& = \{ x \mid x \in A \land
|
|
x \in \{ y \mid y \in B \land y \in C \}\} \\
|
|
& = \{ x \mid x \in A \land (x \in B \land x \in C) \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \land x \in C \} \\
|
|
& = \{ x \mid x \in \{ y \mid y \in A \land y \in B \} \land
|
|
x \in C \} \\
|
|
& = \{ x \mid x \in (A \cap B) \land x \in C \} \\
|
|
& = (A \cap B) \cap C.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Distributive Laws}}%
|
|
\label{sub:distributive-laws}
|
|
|
|
For any sets $A$, $B$, and $C$,
|
|
\begin{align*}
|
|
A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\
|
|
A \cup (B \cap C) & = (A \cup B) \cap (A \cup C)
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.inter\_distrib\_left}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.union\_distrib\_left}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$
|
|
\item $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By the definition of the union and intersection of sets,
|
|
\begin{align*}
|
|
A \cap (B \cup C)
|
|
& = \{ x \mid x \in A \land x \in B \cup C \} \\
|
|
& = \{ x \mid x \in A \land
|
|
x \in \{ y \mid y \in B \lor y \in C \}\} \\
|
|
& = \{ x \mid x \in A \land (x \in B \lor x \in C) \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \lor
|
|
(x \in A \land x \in C) \} \\
|
|
& = \{ x \mid x \in A \cap B \lor x \in A \cap C \} \\
|
|
& = (A \cap B) \cup (A \cap C).
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By the definition of the union and intersection of sets,
|
|
\begin{align*}
|
|
A \cup (B \cap C)
|
|
& = \{ x \mid x \in A \lor x \in B \cap C \} \\
|
|
& = \{ x \mid x \in A \lor
|
|
x \in \{ y \mid y \in B \land y \in C \}\} \\
|
|
& = \{ x \mid x \in A \lor (x \in B \land x \in C) \} \\
|
|
& = \{ x \mid (x \in A \lor x \in B) \land
|
|
(x \in A \lor x \in C) \} \\
|
|
& = \{ x \mid x \in A \cup B \land x \in A \cup C \} \\
|
|
& = (A \cup B) \cap (A \cup C).
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{De Morgan's Laws}}%
|
|
\label{sub:de-morgans-laws}
|
|
|
|
For any sets $A$, $B$, and $C$,
|
|
\begin{align*}
|
|
C - (A \cup B) & = (C - A) \cap (C - B) \\
|
|
C - (A \cap B) & = (C - A) \cup (C - B)
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.diff\_inter\_diff}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.diff\_inter}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $C - (A \cup B) = (C - A) \cap (C - B)$
|
|
\item $C - (A \cap B) = (C - A) \cup (C - B)$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By definition of the union, intersection, and relative complements of sets,
|
|
\begin{align*}
|
|
C - (A \cup B)
|
|
& = \{ x \mid x \in C \land x \not\in A \cup B \} \\
|
|
& = \{ x \mid x \in C \land
|
|
x \not\in \{ y \mid y \in A \lor y \in B \}\} \\
|
|
& = \{ x \mid x \in C \land \neg(x \in A \lor x \in B) \} \\
|
|
& = \{ x \mid x \in C \land (x \not\in A \land x \not\in B) \} \\
|
|
& = \{ x \mid (x \in C \land x \not\in A) \land
|
|
(x \in C \land x \not\in B) \} \\
|
|
& = \{ x \mid x \in (C - A) \land x \in (C - B) \} \\
|
|
& = (C - A) \cap (C - B).
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By definition of the union, intersection, and relative complements of sets,
|
|
\begin{align*}
|
|
C - (A \cap B)
|
|
& = \{ x \mid x \in C \land x \not\in A \cap B \} \\
|
|
& = \{ x \mid x \in C \land
|
|
x \not\in \{ y \mid y \in A \land y \in B \}\} \\
|
|
& = \{ x \mid x \in C \land \neg(x \in A \land x \in B) \} \\
|
|
& = \{ x \mid x \in C \land (x \not\in A \lor x \not\in B) \} \\
|
|
& = \{ x \mid (x \in C \land x \not\in A) \lor
|
|
(x \in C \land x \not\in B) \} \\
|
|
& = \{ x \mid x \in (C - A) \lor x \in (C - B) \} \\
|
|
& = (C - A) \cup (C - B).
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{%
|
|
Identities Involving \texorpdfstring{$\emptyset$}{the Empty Set}}}%
|
|
\label{sub:identitives-involving-empty-set}
|
|
|
|
For any set $A$,
|
|
\begin{align*}
|
|
A \cup \emptyset & = A \\
|
|
A \cap \emptyset & = \emptyset \\
|
|
A \cap (C - A) & = \emptyset
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.union\_empty}
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.inter\_empty}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.inter\_diff\_self}
|
|
|
|
\noindent Let $A$ be an arbitrary set.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \cup \emptyset = A$
|
|
\item $A \cap \emptyset = \emptyset$
|
|
\item $A \cap (C - A) = \emptyset$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By definition of the emptyset and union of sets,
|
|
\begin{align*}
|
|
A \cup \emptyset
|
|
& = \{ x \mid x \in A \lor x \in \emptyset \} \\
|
|
& = \{ x \mid x \in A \lor F \} \\
|
|
& = \{ x \mid x \in A \} \\
|
|
& = A.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By definition of the emptyset and intersection of sets,
|
|
\begin{align*}
|
|
A \cap \emptyset
|
|
& = \{ x \mid x \in A \land x \in \emptyset \} \\
|
|
& = \{ x \mid x \in A \land F \} \\
|
|
& = \{ x \mid F \} \\
|
|
& = \{ x \mid x \neq x \} \\
|
|
& = \emptyset.
|
|
\end{align*}
|
|
|
|
\paragraph{(iii)}%
|
|
|
|
By definition of the emptyset, and the intersection and relative complement
|
|
of sets,
|
|
\begin{align*}
|
|
A \cap (C - A)
|
|
& = \{ x \mid x \in A \land x \in C - A \} \\
|
|
& = \{ x \mid x \in A \land
|
|
x \in \{ y \mid y \in C \land y \not\in A \}\} \\
|
|
& = \{ x \mid x \in A \land (x \in C \land x \not\in A) \} \\
|
|
& = \{ x \mid x \in C \land F \} \\
|
|
& = \{ x \mid F \} \\
|
|
& = \{ x \mid x \neq x \} \\
|
|
& = \emptyset.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Monotonicity}}%
|
|
\label{sub:monotonicity}
|
|
|
|
For any sets $A$, $B$, and $C$,
|
|
\begin{align*}
|
|
A \subseteq B & \Rightarrow A \cup C \subseteq B \cup C \\
|
|
A \subseteq B & \Rightarrow A \cap C \subseteq B \cap C \\
|
|
A \subseteq B & \Rightarrow \bigcup A \subseteq \bigcup B
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.union\_subset\_union\_left}
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.inter\_subset\_inter\_left}
|
|
|
|
\lean{Mathlib/Data/Set/Lattice}{Set.sUnion\_mono}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be arbitrary sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$
|
|
\item $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$
|
|
\item $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
Suppose $A \subseteq B$.
|
|
Let $x \in A \cup C$.
|
|
There are two cases to consider.
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Suppose $x \in A$.
|
|
Then, by definition of the subset, $x \in B$.
|
|
Therefore $x \in B \cup C$.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Suppose $x \in C$.
|
|
Then $x$ is trivially a member of $B \cup C$.
|
|
|
|
\subparagraph{Conclusion}%
|
|
|
|
Since these cases are exhaustive and both imply $x \in B \cup C$, it
|
|
follows $A \cup C \subseteq B \cup C$.
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
Suppose $A \subseteq B$.
|
|
Let $x \in A \cap C$.
|
|
Then, by definition of the intersection of sets, $x \in A$ and $x \in C$.
|
|
By definition of the subset, $x \in A$ implies $x \in B$.
|
|
Therefore $x \in B$ and $x \in C$.
|
|
That is, $x \in B \cap C$.
|
|
Since this holds for arbitrary $x \in A \cap C$, it follows
|
|
$A \cap C \subseteq B \cap C$.
|
|
|
|
\paragraph{(iii)}%
|
|
|
|
Suppose $A \subseteq B$.
|
|
Let $x \in \bigcup A$.
|
|
Then, by definition of the union of sets, there exists some $b \in A$ such
|
|
that $x \in b$.
|
|
By definition of the subset, $b \in B$ as well.
|
|
Another application of the definition of the union of sets immediately
|
|
implies that $x$ is a member of $\bigcup B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Anti-monotonicity}}%
|
|
\label{sub:anti-monotonicity}
|
|
|
|
For any sets $A$, $B$, and $C$,
|
|
\begin{align*}
|
|
A \subseteq B & \Rightarrow C - B \subseteq C - A \\
|
|
\emptyset \neq A \subseteq B & \Rightarrow \bigcap B \subseteq \bigcap A.
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Mathlib/Data/Set/Basic}{Set.diff\_subset\_diff\_right}
|
|
|
|
\lean{Mathlib/Data/Set/Lattice}{Set.sInter\_subset\_sInter}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be arbitrary sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A \subseteq B \Rightarrow C - B \subseteq C - A$
|
|
\item $\emptyset \neq A \subseteq B \Rightarrow
|
|
\bigcap B \subseteq \bigcap A$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
Suppose $A \subseteq B$.
|
|
Let $x \in C - B$.
|
|
By definition of the relative complement, $x \in C$ and $x \not\in B$.
|
|
Then $x$ cannot be a member of $A$, since otherwise this would contradict
|
|
our subset hypothesis.
|
|
That is, $x \in C$ and $x \not\in A$.
|
|
Therefore $x \in C - A$.
|
|
Since this holds for arbitrary $x \in C - B$, it follows that
|
|
$C - B \subseteq C - A$.
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
Suppose $A \neq \emptyset$ and $A \subseteq B$.
|
|
Then $B \neq \emptyset$.
|
|
Let $x \in \bigcap B$.
|
|
By definition of the intersection of sets, for all $b \in B$, $x \in b$.
|
|
But then, by definition of the subset, for all $a \in A$, $x \in a$.
|
|
Therefore $x \in \bigcap A$.
|
|
Since this holds for arbitrary $x \in \bigcap B$, it follows that
|
|
$\bigcap B \subseteq \bigcap A$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{General Distributive Laws}}%
|
|
\label{sub:general-distributive-laws}
|
|
|
|
For any sets $A$ and $\mathscr{B}$,
|
|
\begin{align*}
|
|
A \cup \bigcap \mathscr{B} & =
|
|
\bigcap\; \{ A \cup X \mid X \in \mathscr{B} \}
|
|
\quad\text{for}\quad \mathscr{B} \neq \emptyset \\
|
|
A \cap \bigcup \mathscr{B} & =
|
|
\bigcup\; \{ A \cap X \mid X \in \mathscr{B} \}
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
Let $A$ and $\mathscr{B}$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item For $\mathscr{B} \neq \emptyset$,
|
|
$A \cup \bigcap \mathscr{B} =
|
|
\bigcap\; \{ A \cup X \mid X \in \mathscr{B} \}$.
|
|
\item $A \cap \bigcup \mathscr{B} =
|
|
\bigcup\; \{ A \cap X \mid X \in \mathscr{B} \}$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
Suppose $\mathscr{B}$ is nonempty.
|
|
Then $\bigcap \mathscr{B}$ is defined.
|
|
By definition of the union and intersection of sets,
|
|
\begin{align*}
|
|
A \cup \bigcap \mathscr{B}
|
|
& = \{ x \mid x \in A \lor x \in \bigcap \mathscr{B} \} \\
|
|
& = \{ x \mid x \in A \lor
|
|
x \in \{ y \mid (\forall b \in \mathscr{B}), y \in b \}\} \\
|
|
& = \{ x \mid x \in A \lor (\forall b \in \mathscr{B}), x \in b \} \\
|
|
& = \{ x \mid \forall b \in \mathscr{B}, x \in A \lor x \in b \} \\
|
|
& = \{ x \mid \forall b \in \mathscr{B}, x \in A \cup b \} \\
|
|
& = \{ x \mid
|
|
x \in \bigcap\; \{ A \cup X \mid X \in \mathscr{B} \}\} \\
|
|
& = \bigcap\; \{ A \cup X \mid X \in \mathscr{B} \}.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By definition of the intersection and union of sets,
|
|
\begin{align*}
|
|
A \cap \bigcup \mathscr{B}
|
|
& = \{ x \mid x \in A \land x \in \bigcup \mathscr{B} \} \\
|
|
& = \{ x \mid x \in A \land
|
|
x \in \{ y \mid (\exists b \in \mathscr{B}), y \in b \}\} \\
|
|
& = \{ x \mid x \in A \land (\exists b \in \mathscr{B}), x \in b \} \\
|
|
& = \{ x \mid \exists b \in \mathscr{B}, x \in A \land x \in b \} \\
|
|
& = \{ x \mid \exists b \in \mathscr{B} x \in A \cap b \} \\
|
|
& = \{ x \mid
|
|
x \in \bigcup\; \{ A \cap X \mid X \in \mathscr{B} \}\} \\
|
|
& = \bigcup\; \{ A \cap X \mid X \in \mathscr{B} \}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{General De Morgan's Laws}}%
|
|
\label{sub:general-de-morgans-laws}
|
|
|
|
For any set $C$ and $\mathscr{A} \neq \emptyset$,
|
|
\begin{align*}
|
|
C - \bigcup \mathscr{A} & = \bigcap\; \{ C - X \mid X \in \mathscr{A} \} \\
|
|
C - \bigcap \mathscr{A} & = \bigcup\; \{ C - X \mid X \in \mathscr{A} \}
|
|
\end{align*}
|
|
|
|
\begin{proof}
|
|
|
|
Let $C$ and $\mathscr{A}$ be sets such that $\mathscr{A} \neq \emptyset$.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $C - \bigcup \mathscr{A} =
|
|
\bigcap\; \{ C - X \mid X \in \mathscr{A} \}$
|
|
\item $C - \bigcap \mathscr{A} =
|
|
\bigcup\; \{ C - X \mid X \in \mathscr{A} \}$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By definition of the relative complement, union, and intersection of sets,
|
|
\begin{align*}
|
|
C - \bigcup \mathscr{A}
|
|
& = \{ x \mid x \in C \land x \not\in \bigcup \mathscr{A} \} \\
|
|
& = \{ x \mid x \in C \land
|
|
x \not\in \{ y \mid (\exists b \in \mathscr{A}) y \in b \}\} \\
|
|
& = \{ x \mid x \in C \land
|
|
\neg(\exists b \in \mathscr{A}, x \in b) \} \\
|
|
& = \{ x \mid x \in C \land
|
|
(\forall b \in \mathscr{A}, x \not\in b) \} \\
|
|
& = \{ x \mid
|
|
\forall b \in \mathscr{A}, x \in C \land x \not\in b \} \\
|
|
& = \{ x \mid \forall b \in \mathscr{A}, x \in C - b \} \\
|
|
& = \{ x \mid x \in \bigcap\; \{ C - X \mid X \in \mathscr{A} \} \\
|
|
& = \bigcap\; \{ C - X \mid X \in \mathscr{A} \}.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By definition of the relative complement, union, and intersection of sets,
|
|
\begin{align*}
|
|
C - \bigcap \mathscr{A}
|
|
& = \{ x \mid x \in C \land x \not\in \bigcap \mathscr{A} \} \\
|
|
& = \{ x \mid x \in C \land
|
|
x \not\in \{ y \mid (\forall b \in \mathscr{A}) y \in b \}\} \\
|
|
& = \{ x \mid x \in C \land
|
|
\neg(\forall b \in \mathscr{A}, x \in b) \} \\
|
|
& = \{ x \mid x \in C \land
|
|
\exists b \in \mathscr{A}, x \not\in b \} \\
|
|
& = \{ x \mid
|
|
\exists b \in \mathscr{A}, x \in C \land x \not\in b \} \\
|
|
& = \{ x \mid \exists b \in \mathscr{A}, x \in C - b \} \\
|
|
& = \{ x \mid x \in \bigcup\; \{ C - X \mid X \in \mathscr{A} \} \} \\
|
|
& = \bigcup\; \{ C - X \mid X \in \mathscr{A} \}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{%
|
|
\texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
|
|
\label{sub:intersection-difference-associativity}
|
|
|
|
Let $A$, $B$, and $C$ be sets.
|
|
Then $A \cap (B - C) = (A \cap B) - C$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.inter\_diff\_assoc}
|
|
|
|
Let $A$, $B$, and $C$ be sets.
|
|
By definition of the intersection and relative complement of sets,
|
|
\begin{align*}
|
|
A \cap (B - C)
|
|
& = \{ x \mid x \in A \land x \in B - C \} \\
|
|
& = \{ x \mid x \in A \land (x \in B \land x \not\in C) \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \land x \not\in C \} \\
|
|
& = \{ x \mid x \in A \cap B \land x \not \in C \} \\
|
|
& = (A \cap B) - C.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Nonmembership of Symmetric Difference}}
|
|
\label{sub:nonmembership-symmetric-difference}
|
|
|
|
Let $A$ and $B$ be sets. $x \not\in A + B$ if and only if either
|
|
$x \in A \cap B$ or $x \not\in A \cup B$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Common/Set/Basic}{Set.not\_mem\_symm\_diff\_inter\_or\_not\_union}
|
|
|
|
By definition of the \nameref{ref:symmetric-difference},
|
|
\begin{align*}
|
|
x \not\in A + B
|
|
& = \neg(x \in A + B) \\
|
|
& = \neg[x \in (A - B) \cup (B - A)] \\
|
|
& = \neg[x \in (A - B) \lor x \in (B - A)] \\
|
|
& = \neg[(x \in A \land x \not\in B) \lor
|
|
(x \in B \land x \not\in A)] \\
|
|
& = \neg(x \in A \land x \not\in B) \land
|
|
\neg(x \in B \land x \not\in A) \\
|
|
& = (x \not\in A \lor x \in B) \land (x \not\in B \lor x \in A) \\
|
|
& = ((x \not\in A \lor x \in B) \land x \not\in B) \lor
|
|
((x \not\in A \lor x \in B) \land x \in A) \\
|
|
& = (x \not\in A \land x \not\in B) \lor (x \in B \land x \in A) \\
|
|
& = \neg(x \in A \lor x \in B) \lor (x \in B \land x \in A) \\
|
|
& = x \not\in A \cup B \text{ or } x \in A \cap B.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\section{Exercises 4}%
|
|
\label{sec:exercises-4}
|
|
|
|
\subsection{\verified{Exercise 4.11}}%
|
|
\label{sub:exercise-4.11}
|
|
|
|
Show that for any sets $A$ and $B$,
|
|
$$A = (A \cap B) \cup (A - B) \quad\text{and}\quad
|
|
A \cup (B - A) = A \cup B.$$
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_11\_i}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_11\_ii}
|
|
|
|
\noindent Let $A$ and $B$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A = (A \cap B) \cup (A - B)$
|
|
\item $A \cup (B - A) = A \cup B$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
|
|
By definition of the intersection, union, and relative complements of sets,
|
|
\begin{align*}
|
|
(A \cap B) \cup (A - B)
|
|
& = \{ x \mid x \in A \cap B \lor x \in A - B \} \\
|
|
& = \{ x \mid x \in \{ y \mid y \in A \land y \in B \} \lor
|
|
x \in A - B \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \lor x \in A - B \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \lor
|
|
x \in \{ y \mid y \in A \land y \not\in B \} \} \\
|
|
& = \{ x \mid (x \in A \land x \in B) \lor
|
|
(x \in A \land x \not\in B) \} \\
|
|
& = \{ x \mid x \in A \lor (x \in B \land x \not\in B) \} \\
|
|
& = \{ x \mid x \in A \lor F \} \\
|
|
& = \{ x \mid x \in A \} \\
|
|
& = A.
|
|
\end{align*}
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By definition of the union and relative complements of sets,
|
|
\begin{align*}
|
|
A \cup (B - A)
|
|
& = \{ x \mid x \in A \lor x \in B - A \} \\
|
|
& = \{ x \mid x \in A \lor
|
|
x \in \{ y \mid y \in B \land y \not\in A \} \} \\
|
|
& = \{ x \mid x \in A \lor (x \in B \land x \not\in A) \} \\
|
|
& = \{ x \mid (x \in A \lor x \in B) \land
|
|
(x \in A \lor x \not\in A) \} \\
|
|
& = \{ x \mid (x \in A \lor x \in B) \land T \} \\
|
|
& = \{ x \mid x \in A \lor x \in B \} \\
|
|
& = \{ x \mid x \in A \cup B \} \\
|
|
& = A \cup B.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.12}}%
|
|
\label{sub:exercise-4.12}
|
|
|
|
Verify the following identity (one of De Morgan's laws):
|
|
$$C - (A \cap B) = (C - A) \cup (C - B).$$
|
|
|
|
\begin{proof}
|
|
|
|
Refer to \nameref{sub:de-morgans-laws}.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.13}}%
|
|
\label{sub:exercise-4.13}
|
|
|
|
Show that if $A \subseteq B$, then $C - B \subseteq C - A$.
|
|
|
|
\begin{proof}
|
|
|
|
Refer to \nameref{sub:anti-monotonicity}.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.14}}%
|
|
\label{sub:exercise-4.14}
|
|
|
|
Show by example that for some sets $A$, $B$, and $C$, the set $A - (B - C)$ is
|
|
different from $(A - B) - C$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_14}
|
|
|
|
Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$, and $C = \{3, 4, 5\}$.
|
|
Then
|
|
\begin{align*}
|
|
A - (B - C)
|
|
& = \{1, 2, 3\} - (\{2, 3, 4\} - \{3, 4, 5\}) \\
|
|
& = \{1, 2, 3\} - \{2\} \\
|
|
& = \{1, 3\}
|
|
\end{align*}
|
|
but
|
|
\begin{align*}
|
|
(A - B) - C
|
|
& = (\{1, 2, 3\} - \{2, 3, 4\}) - \{3, 4, 5\} \\
|
|
& = \{1\} - \{3, 4, 5\} \\
|
|
& = \{1\}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.15a}}%
|
|
\label{sub:exercise-4.15a}
|
|
|
|
Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.inter\_symmDiff\_distrib\_left}
|
|
|
|
By definition of the intersection, \nameref{ref:symmetric-difference}, and
|
|
relative complement of sets,
|
|
\begin{align*}
|
|
(A & \cap B) + (A \cap C) \\
|
|
& = [(A \cap B) - (A \cap C)] \cup [(A \cap C) - (A \cap B)] \\
|
|
& = [(A \cap B) - A] \\
|
|
& \qquad \cup [(A \cap B) - C] \\
|
|
& \qquad \cup [(A \cap C) - A] \\
|
|
& \qquad \cup [(A \cap C) - B]
|
|
& \textref{sub:de-morgans-laws} \\
|
|
& = [A \cap (B - A)] \\
|
|
& \qquad \cup [A \cap (B - C)] \\
|
|
& \qquad \cup [A \cap (C - A)] \\
|
|
& \qquad \cup [A \cap (C - B)]
|
|
& \textref{sub:intersection-difference-associativity} \\
|
|
& = \emptyset \\
|
|
& \qquad \cup [A \cap (B - C)] \\
|
|
& \qquad \cup \emptyset \\
|
|
& \qquad \cup [A \cap (C - B)]
|
|
& \textref{sub:identitives-involving-empty-set} \\
|
|
& = [A \cap (B - C)] \cup [A \cap (C - B)] \\
|
|
& = A \cap [(B - C) \cup (C - B)]
|
|
& \textref{sub:distributive-laws} \\
|
|
& = A \cap (B + C).
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.15b}}%
|
|
\label{sub:exercise-4.15b}
|
|
|
|
Show that $A + (B + C) = (A + B) + C$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Mathlib/Data/Set/Basic}{Set.symmDiff\_assoc}
|
|
|
|
\noindent Let $A$, $B$, and $C$ be sets.
|
|
We prove that
|
|
\begin{enumerate}[(i)]
|
|
\item $A + (B + C) \subseteq (A + B) + C$
|
|
\item $(A + B) + C \subseteq A + (B + C)$
|
|
\end{enumerate}
|
|
|
|
\paragraph{(i)}%
|
|
\label{par:exercise-4.15b-i}
|
|
|
|
Let $x \in A + (B + C)$.
|
|
Then $x$ is in $A$ or in $B + C$, but not both.
|
|
There are two cases to consider:
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Suppose $x \in A$ and $x \not\in B + C$.
|
|
Then, by \nameref{sub:nonmembership-symmetric-difference},
|
|
(a) $x \in B \cap C$ or (b) $x \not\in B \cup C$.
|
|
Suppose (a) was true.
|
|
That is, $x \in B$ and $x \in C$.
|
|
Since $x$ is a member of $A$ and $B$, $x \not\in (A + B)$.
|
|
Since $x$ is not a member of $A + B$ but is a member of $C$,
|
|
$x \in (A + B) + C$.
|
|
Now suppose (b) was true.
|
|
That is, $x \not\in B$ and $x \not\in C$.
|
|
Since $x$ is a member of $A$ but not $B$, $x \in (A + B)$.
|
|
Since $x$ is a member of $A + B$ but not $C$, $x \in (A + B) + C$.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Suppose $x \in B + C$ and $x \not\in A$.
|
|
Then (a) $x \in B$ or (b) $x \in C$ but not both.
|
|
Suppose (a) was true.
|
|
That is, $x \in B$ and $x \not\in C$.
|
|
Since $x$ is not a member of $A$ and is a member of $B$, $x \in A + B$.
|
|
Since $x$ is a member of $A + B$ but not $C$, $x \in (A + B) + C$.
|
|
Now suppose (b) was true.
|
|
That is, $x \not\in B$ and $x \in C$.
|
|
Since $x$ is not a member of $A$ nor a member of $B$, $x \not\in A + B$.
|
|
Since $x$ is not a member of $A + B$ but is a member of $C$,
|
|
$x \in (A + B) + C$.
|
|
|
|
\paragraph{(ii)}%
|
|
\label{par:exercise-4.15b-ii}
|
|
|
|
Let $x \in (A + B) + C$.
|
|
Then $x$ is in $A + B$ or in $C$, but not both.
|
|
There are two cases to consider:
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Suppose $x \in A + B$ and $x \not\in C$.
|
|
Then (a) $x \in A$ or (b) $x \in B$ but not both.
|
|
Suppose (a) was true.
|
|
That is, $x \in A$ and $x \not\in B$.
|
|
Since $x$ is not a member of $B$ nor $C$, $x \not\in B + C$.
|
|
Since $x$ is not a member of $B + C$ but is a member of $A$,
|
|
$x \in A + (B + C)$.
|
|
Now Suppose (b) was true.
|
|
That is, $x \not\in A$ and $x \in B$.
|
|
Since $x$ is a member of $B$ and not of $C$, then $x \in B + C$.
|
|
Since $x$ is a member of $B + C$ and not of $A$, $x \in A + (B + C)$.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Suppose $x \not\in A + B$ and $x \in C$.
|
|
Then, by \nameref{sub:nonmembership-symmetric-difference},
|
|
(a) $x \in A \cap B$ or (b) $x \not\in A \cup B$.
|
|
Suppose (a) was true.
|
|
That is, $x \in A \land x \in B$.
|
|
Since $x$ is a member of $B$ and $C$, $x \not\in B + C$.
|
|
Since $x$ is not a member of $B + C$ but is a member of $A$,
|
|
$x \in A + (B + C)$.
|
|
Now suppose (b) was true.
|
|
That is, $x \not\in A$ and $x \not\in B$.
|
|
Since $x$ is not a member of $B$ but is a member of $C$, $x \in B + C$.
|
|
Since $x$ is a member of $B + C$ but not of $A$, $x \in A + (B + C)$.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
In both \nameref{par:exercise-4.15b-i} and \nameref{par:exercise-4.15b-ii},
|
|
the subcases are exhaustive and prove the desired subset relation.
|
|
Therefore $A + (B + C) = (A + B) + C$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.16}}%
|
|
\label{sub:exercise-4.16}
|
|
|
|
Simplify:
|
|
$$[(A \cup B \cup C) \cap (A \cup B)] - [(A \cup (B - C)) \cap A].$$
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_16}
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets.
|
|
Then
|
|
\begin{align*}
|
|
[(A \cup B \cup C) \cap (A \cup B)] & - [(A \cup (B - C)) \cap A] \\
|
|
& = [A \cup B] - [A] \\
|
|
& = \{ x \mid x \in (A \cup B) \land x \not\in A \} \\
|
|
& = \{ x \mid x \in \{ y \mid y \in A \lor y \in B \} \land x \not\in A \} \\
|
|
& = \{ x \mid (x \in A \lor x \in B) \land x \not\in A \} \\\
|
|
& = \{ x \mid (x \in A \land x \not\in A) \lor (x \in B \land x \not\in A) \} \\
|
|
& = \{ x \mid F \lor (X \in B \land x \not\in A) \} \\
|
|
& = \{ x \mid x \in B \land x \not\in A \} \\
|
|
& = B - A.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.17}}%
|
|
\label{sub:exercise-4.17}
|
|
|
|
Show that the following four conditions are equivalent.
|
|
|
|
\begin{enumerate}[(a)]
|
|
\item $A \subseteq B$,
|
|
\item $A - B = \emptyset$,
|
|
\item $A \cup B = B$,
|
|
\item $A \cap B = A$.
|
|
\end{enumerate}
|
|
|
|
\begin{proof}
|
|
|
|
\statementpadding
|
|
|
|
\lean*{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_17\_i}
|
|
|
|
\lean*{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_17\_ii}
|
|
|
|
\lean*{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_17\_iii}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_17\_iv}
|
|
|
|
Let $A$ and $B$ be arbitrary sets.
|
|
We show that (i) $(a) \Rightarrow (b)$, (ii) $(b) \Rightarrow (c)$, (iii)
|
|
$(c) \Rightarrow (d)$, and (iv) $(d) \Rightarrow (a)$.
|
|
|
|
\paragraph{(i)}%
|
|
|
|
Suppose $A \subseteq B$.
|
|
That is, $\forall t, t \in A \Rightarrow t \in B$.
|
|
Then there is no element such that $t \in A$ and $t \not\in B$.
|
|
By definition of the relative complement, this immediately implies
|
|
$A - B = \emptyset$.
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
Suppose $A - B = \emptyset$.
|
|
By definition of the relative complement,
|
|
$$A - B = \emptyset = \{ x \mid x \in A \land x \not\in B \}.$$
|
|
Then, for all $t$,
|
|
$\neg(t \in A \land t \not\in B) = t \not\in A \lor t \in B$.
|
|
This implies, by definition of the subset, that $A \subseteq B$.
|
|
It then immediately follows that $A \cup B = B$.
|
|
|
|
\paragraph{(iii)}%
|
|
|
|
Suppose $A \cup B = B$.
|
|
Then there is no member of $A$ that is not a member of $B$.
|
|
In other words, $A \subseteq B$.
|
|
This immediately implies $A \cap B = A$.
|
|
|
|
\paragraph{(iv)}%
|
|
|
|
Suppose $A \cap B = A$.
|
|
Then every member of $A$ is a member of $B$.
|
|
This immediately implies $A \subseteq B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 4.18}}%
|
|
\label{sub:exercise-4.18}
|
|
|
|
Assume that $A$ and $B$ are subsets of $S$.
|
|
List all of the different sets that can be made from these three by use of the
|
|
binary operations $\cup$, $\cap$, and $-$.
|
|
|
|
\begin{proof}
|
|
|
|
We can reason about this diagrammatically:
|
|
|
|
\begin{figure}[ht]
|
|
\includegraphics[width=0.6\textwidth]{venn-diagram}
|
|
\centering
|
|
\end{figure}
|
|
|
|
In the above diagram, we assume the left circle corresponds to set $A$ and the
|
|
right circle corresponds to $B$.
|
|
The the possible sets we can make via the specified operators are:
|
|
|
|
\begin{itemize}
|
|
\item $A - B$, the left circle excluding the overlapping region.
|
|
\item $A \cap B$, the overlapping region.
|
|
\item $B - A$, the right circle excluding the overlapping region.
|
|
\item $(A \cup B) \cap A$, the left circle.
|
|
\item $(A \cup B) \cap B$, the right circle.
|
|
\item $(A - B) \cup (B - A)$, the symmetric difference.
|
|
\item $A \cup B$, the entire diagram.
|
|
\end{itemize}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.19}}%
|
|
\label{sub:exercise-4.19}
|
|
|
|
Is $\powerset{(A - B)}$ always equal to $\powerset{A} - \powerset{B}$?
|
|
Is it ever equal to $\powerset{A} - \powerset{B}$?
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_19}
|
|
|
|
Let $A$ and $B$ be arbitrary sets.
|
|
We show (i) that $\emptyset \in \powerset{(A - B})$ and (ii)
|
|
$\emptyset \not\in \powerset{A} - \powerset{B}$.
|
|
|
|
\paragraph{(i)}%
|
|
\label{par:exercise-4.19-i}
|
|
|
|
By definition of the \nameref{ref:power-set},
|
|
$$\powerset{(A - B)} = \{ x \mid x \subseteq A - B \}.$$
|
|
But $\emptyset$ is a subset of \textit{every} set.
|
|
Thus $\emptyset \in \powerset{(A - B)}$.
|
|
|
|
\paragraph{(ii)}%
|
|
|
|
By the same reasoning found in \nameref{par:exercise-4.19-i},
|
|
$\emptyset \in \powerset{A}$ and $\emptyset \in \powerset{B}$.
|
|
But then, by definition of the relative complement,
|
|
$\emptyset \not\in \powerset{A} - \powerset{B}$.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
By the \nameref{ref:extensionality-axiom}, the two sets are never equal.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.20}}%
|
|
\label{sub:exercise-4.20}
|
|
|
|
Let $A$, $B$, and $C$ be sets such that $A \cup B = A \cup C$ and
|
|
$A \cap B = A \cap C$.
|
|
Show that $B = C$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_20}
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets.
|
|
By the \nameref{ref:extensionality-axiom}, $B = C$ if and only if for all sets
|
|
$x$, $x \in B \iff x \in C$.
|
|
We prove both directions of this biconditional.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose $x \in B$.
|
|
Then there are two cases to consider:
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Assume $x \in A$.
|
|
Then $x \in A \cap B$.
|
|
By hypothesis, $A \cap B = A \cap C$.
|
|
Thus $x \in A \cap C$ immediately implying $x \in C$.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Assume $x \not\in A$.
|
|
Then $x \in A \cup B$.
|
|
By hypothesis, $A \cup B = A \cup C$.
|
|
Thus $x \in A \cup C$.
|
|
Since $x \not\in A$, it follows $x \in C$.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
Suppose $x \in C$.
|
|
Then there are two cases to consider:
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Assume $x \in A$.
|
|
Then $x \in A \cap C$.
|
|
By hypothesis, $A \cap B = A \cap C$.
|
|
Thus $x \in A \cap B$, immediately implying $x \in B$.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Assume $x \not\in A$.
|
|
Then $x \in A \cup C$.
|
|
By hypothesis, $A \cup B = A \cup C$.
|
|
Thus $x \in A \cup B$.
|
|
Since $x \not\in A$, it follows $x \in B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.21}}%
|
|
\label{sub:exercise-4.21}
|
|
|
|
Show that $\bigcup (A \cup B) = \bigcup A \cup \bigcup B$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_21}
|
|
|
|
Let $A$ and $B$ be arbitrary sets.
|
|
By the \nameref{ref:extensionality-axiom}, the specified equality holds if and
|
|
only if for all sets $x$,
|
|
$$x \in \bigcup (A \cup B) \iff x \in \bigcup A \cup \bigcup B.$$
|
|
We prove both directions of this biconditional.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose $x \in \bigcup (A \cup B)$.
|
|
By definition of the union of sets, there exists some $b \in A \cup B$ such
|
|
that $x \in b$.
|
|
If $b \in A$, then $x \in \bigcup A$ and $x \in \bigcup A \cup \bigcup B$.
|
|
Alternatively, if $b \in B$, then $x \in \bigcup B$ and
|
|
$x \in \bigcup A \cup \bigcup B$.
|
|
Regardless, $x$ is in the target set.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
Suppose $x \in \bigcup A \cup \bigcup B$.
|
|
Then $x \in \bigcup A$ or $x \in \bigcup B$.
|
|
WLOG, suppose $x \in \bigcup A$.
|
|
By definition of the union of sets, there exists some $b \in A$ such that
|
|
$x \in b$.
|
|
But then $b \in A \cup B$ meaning $x$ is also a member of
|
|
$\bigcup (A \cup B)$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.22}}%
|
|
\label{sub:exercise-4.22}
|
|
|
|
Show that if $A$ and $B$ are nonempty sets, then
|
|
$\bigcap (A \cup B) = \bigcap A \cap \bigcap B$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_22}
|
|
|
|
Let $A$ and $B$ be arbitrary, nonempty sets.
|
|
By the \nameref{ref:extensionality-axiom}, the specified equality holds if and
|
|
only if for all sets $x$,
|
|
\begin{equation}
|
|
\label{sub:exercise-4.22-eq1}
|
|
x \in \bigcap (A \cup B) \iff x \in \bigcap A \cap \bigcap B.
|
|
\end{equation}
|
|
We prove both directions of this biconditional.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose $x \in \bigcap (A \cup B)$.
|
|
Then for all $b \in A \cup B$, $x \in B$.
|
|
In other words, for every member $b_1$ of $A$ and every member $b_2$ of $B$,
|
|
$x$ is a member of both $b_1$ and $b_2$.
|
|
But that implies $x \in \bigcap A$ and $x \in \bigcap B$.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
Suppose $x \in \bigcap A \cap \bigcap B$.
|
|
That is, $x \in \bigcap A$ and $x \in \bigcap B$.
|
|
By definition of the intersection of sets, forall sets $b$, if $b \in A$,
|
|
then $x \in b$.
|
|
Likewise, if $b \in B$, then $x \in b$.
|
|
In other words, if $b$ is a member of either $A$ or $B$, $x \in b$.
|
|
That immediately implies $x \in \bigcap (A \cup B$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 4.23}}%
|
|
\label{sub:exercise-4.23}
|
|
|
|
Show that if $\mathscr{B}$ is nonempty, then
|
|
$A \cup \bigcap \mathscr{B} = \bigcap\; \{A \cup X \mid X \in \mathscr{B} \}$.
|
|
|
|
\begin{proof}
|
|
|
|
Refer to \nameref{sub:general-distributive-laws}.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.24a}}%
|
|
\label{sub:exercise-4.24a}
|
|
|
|
Show that if $\mathscr{A}$ is nonempty, then
|
|
$\powerset{\bigcap\mathscr{A}} =
|
|
\bigcap\; \{\powerset{X} \mid X \in \mathscr{A} \}$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_24a}
|
|
|
|
Suppose $\mathscr{A}$ is a nonempty set.
|
|
Then $\bigcap \mathscr{A}$ is well-defined.
|
|
Therefore
|
|
\begin{align*}
|
|
\powerset{\bigcap\mathscr{A}}
|
|
& = \{ x \mid x \subseteq \bigcap \mathscr{A} \}
|
|
& \textref{ref:power-set} \\
|
|
& = \{ x \mid x \subseteq
|
|
\{ y \mid \forall X \in \mathscr{A}, y \in X \} \}
|
|
& \text{def'n intersection} \\
|
|
& = \{ x \mid \forall t \in x,
|
|
t \in \{ y \mid \forall X \in \mathscr{A}, y \in X \} \}
|
|
& \text{def'n subset} \\
|
|
& = \{ x \mid \forall t \in x,
|
|
(\forall X \in \mathscr{A}, t \in X) \} \\
|
|
& = \{ x \mid \forall X \in \mathscr{A},
|
|
(\forall t \in x, t \in X) \} \\
|
|
& = \{ x \mid \forall X \in \mathscr{A}, x \subseteq X \} \\
|
|
& = \{ x \mid \forall X \in \mathscr{A}, x \in \powerset{X} \}
|
|
& \textref{ref:power-set-axiom} \\
|
|
& = \{ x \mid
|
|
\forall t \in \{ \powerset{X} \mid X \in \mathscr{A} \}, x \in t \} \\
|
|
& = \bigcap\; \{\powerset{X} \mid X \in \mathscr{A}\}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.24b}}%
|
|
\label{sub:exercise-4.24b}
|
|
|
|
Show that
|
|
\begin{equation}
|
|
\label{sub:exercise-4.24b-eq1}
|
|
\bigcup\; \{ \powerset{X} \mid X \in \mathscr{A} \} \subseteq
|
|
\powerset{\bigcup\mathscr{A}}.
|
|
\end{equation}
|
|
Under what conditions does equality hold?
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_24b}
|
|
|
|
We first prove \eqref{sub:exercise-4.24b-eq1}.
|
|
Let $x \in \bigcup\; \{ \powerset{X} \mid X \in \mathscr{A} \}$.
|
|
By definition of the union of sets,
|
|
$(\exists X \in \mathscr{A}), x \in \powerset{X}$.
|
|
By definition of the \nameref{ref:power-set}, $x \subseteq X$.
|
|
By \nameref{sub:exercise-3.3}, $X \subseteq \bigcup \mathscr{A}$.
|
|
Therefore $x \subseteq \bigcup \mathscr{A}$, proving
|
|
$x \in \powerset{\mathscr{A}}$ as expected.
|
|
|
|
\suitdivider
|
|
|
|
\noindent
|
|
We show $\powerset{\bigcup A} \subseteq
|
|
\bigcup\;\{ \powerset{X} \mid X \in \mathscr{A} \}$ if and only if
|
|
$\bigcup\mathscr{A} \in \mathscr{A}$.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose $\powerset{\bigcup\mathscr{A}} \subseteq
|
|
\bigcup\;\{ \powerset{X} \mid X \in \mathscr{A} \}$.
|
|
By definition of the \nameref{ref:power-set},
|
|
$\bigcup\mathscr{A} \in \powerset{\bigcup\mathscr{A}}$.
|
|
By hypothesis, $\bigcup\mathscr{A} \in
|
|
\bigcup\;\{ \powerset{X} \mid X \in \mathscr{A} \}$.
|
|
By definition of the union of sets, there exists some $X \in \mathscr{A}$
|
|
such that $\bigcup\mathscr{A} \in \powerset{X}$.
|
|
That is, $\bigcup\mathscr{A} \subseteq X$.
|
|
But $\bigcup\mathscr{A}$ cannot be a proper subset of $X$ since
|
|
$X \in \mathscr{A}$.
|
|
Thus $\bigcup\mathscr{A} = X$.
|
|
This proves $\bigcup\mathscr{A} \in
|
|
\bigcup\;\{ \powerset{X} \mid X \in \mathscr{A} \}$.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
Suppose $\bigcup\mathscr{A} \in A$.
|
|
Let $x \in \powerset{\bigcup\mathscr{A}}$.
|
|
Since $\bigcup\mathscr{A} \in \mathscr{A}$, it immediately follows that
|
|
$x \in \{\powerset{X} \mid X \in \mathscr{A}\}$.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
Equality follows immediately from this fact in conjunction with the proof
|
|
of \eqref{sub:exercise-4.24b-eq1}.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 4.25}}%
|
|
\label{sub:exercise-4.25}
|
|
|
|
Is $A \cup \bigcup \mathscr{B}$ always the same as
|
|
$\bigcup\;\{ A \cup X \mid X \in \mathscr{B} \}$?
|
|
If not, then under what conditions does equality hold?
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_2}
|
|
{Enderton.Set.Chapter\_2.exercise\_4\_25}
|
|
|
|
We prove that
|
|
\begin{equation}
|
|
\label{sub:exercise-4.25-eq1}
|
|
A \cup \bigcup \mathscr{B} =
|
|
\bigcup\;\{ A \cup X \mid X \in \mathscr{B} \}
|
|
\end{equation}
|
|
if and only if $A = \emptyset$ or $\mathscr{B} \neq \emptyset$.
|
|
We prove both directions of this biconditional.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose \eqref{sub:exercise-4.25-eq1} holds true.
|
|
There are two cases to consider:
|
|
|
|
\subparagraph{Case 1}%
|
|
|
|
Suppose $B \neq \emptyset$.
|
|
Then $A = \emptyset \lor \mathscr{B} \neq \emptyset$ holds trivially.
|
|
|
|
\subparagraph{Case 2}%
|
|
|
|
Suppose $B = \emptyset$.
|
|
Then $$A \cup \bigcup \mathscr{B} = A \cup \bigcup \emptyset = A$$ and
|
|
$$
|
|
\bigcup\;\{ A \cup X \mid X \in \mathscr{B} \}
|
|
= \bigcup \emptyset \\
|
|
= \emptyset.
|
|
$$
|
|
Then by hypothesis \eqref{sub:exercise-4.25-eq1}, $A = \emptyset$.
|
|
Then $A = \emptyset \lor \mathscr{B} \neq \emptyset$ holds trivially.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
Suppose $A = \emptyset$ or $\mathscr{B} \neq \emptyset$.
|
|
There are two cases to consider:
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
Suppose $A = \emptyset$.
|
|
Then $A \cup \bigcup \mathscr{B} = \bigcup{\mathscr{B}}$.
|
|
Likewise,
|
|
$$
|
|
\bigcup \{ A \cup X \mid X \in \mathscr{B} \}
|
|
= \bigcup \{ X \mid X \in \mathscr{B} \} \\
|
|
= \bigcup \mathscr{B}.
|
|
$$
|
|
Therefore \eqref{sub:exercise-4.25-eq1} holds.
|
|
|
|
\paragraph{Case 2}%
|
|
|
|
Suppose $B \neq \emptyset$.
|
|
Then
|
|
\begin{align*}
|
|
A \cup \bigcup\mathscr{B}
|
|
& = \{ x \mid x \in A \lor x \in \bigcup\mathscr{B} \} \\
|
|
& = \{ x \mid x \in A \lor (\exists b \in \mathscr{B}) x \in b \} \\
|
|
& = \{ x \mid (\exists b \in \mathscr{B}) x \in A \lor x \in b \} \\
|
|
& = \{ x \mid (\exists b \in \mathscr{B}) x \in A \cup b \} \\
|
|
& = \{ x \mid x \in \bigcup \{ A \cup X \mid X \in \mathscr{B} \} \\
|
|
& = \bigcup \{ A \cup X \mid X \in \mathscr{B} \}.
|
|
\end{align*}
|
|
Therefore \eqref{sub:exercise-4.25-eq1} holds.
|
|
|
|
\end{proof}
|
|
|
|
\chapter{Relations and Functions}%
|
|
\label{chap:relations-functions}
|
|
|
|
\section{Ordered Pairs}%
|
|
\label{sec:ordered-pairs}
|
|
|
|
\subsection{\verified{Theorem 3A}}%
|
|
\label{sub:theorem-3a}
|
|
|
|
\begin{theorem}[3A]
|
|
|
|
For any sets $x$, $y$, $u$, and $v$,
|
|
\begin{equation}
|
|
\label{sub:theorem-3a-eq1}
|
|
\left< u, v \right> = \left< x, y \right> \iff u = x \land v = y.
|
|
\end{equation}
|
|
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Common/Set/OrderedPair}{Set.OrderedPair.ext\_iff}
|
|
|
|
Let $x$, $y$, $u$, and $v$ be arbitrary sets.
|
|
|
|
\paragraph{($\Leftarrow$)}%
|
|
|
|
This follows trivially.
|
|
|
|
\paragraph{($\Rightarrow$)}%
|
|
|
|
Suppose $\left< u, v \right> = \left< x, y \right>$.
|
|
Then, by definition of an \nameref{ref:ordered-pair},
|
|
\begin{equation}
|
|
\label{sub:theorem-3a-eq2}
|
|
\{\{u\}, \{u, v\}\} = \{\{x\}, \{x, y\}\}.
|
|
\end{equation}
|
|
By the \nameref{ref:extensionality-axiom}, it follows
|
|
$\{u\} \in \{\{x\}, \{x, y\}\}$ and
|
|
$\{u, v\} \in \{\{x\}, \{x, y\}\}$.
|
|
That is,
|
|
$$\{u\} = \{x\} \quad\text{or}\quad \{u\} = \{x, y\}$$
|
|
and
|
|
$$\{u, v\} = \{x\} \quad\text{or}\quad \{u, v\} = \{x, y\}.$$
|
|
There are 4 cases to consider:
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
Suppose $\{u\} = \{x\}$ and $\{u, v\} = \{x\}$.
|
|
The former identity implies $u = x$.
|
|
The latter identity implies $u = v = x$.
|
|
Then \eqref{sub:theorem-3a-eq2} simplifies to
|
|
$$\{\{u\}\} = \{\{x\}, \{x, y\}\},$$ meaning $x = y$.
|
|
Thus $v = y$ as well.
|
|
|
|
\paragraph{Case 2}%
|
|
|
|
Suppose $\{u\} = \{x\}$ and $\{u, v\} = \{x, y\}$.
|
|
The former identity implies $u = x$.
|
|
Substituting into the latter identity yields $\{u, v\} = \{u, y\}$.
|
|
This holds if and only if $v = y$.
|
|
|
|
\paragraph{Case 3}%
|
|
|
|
Suppose $\{u\} = \{x, y\}$ and $\{u, v\} = \{x\}$.
|
|
The former identity implies $x = y = u$.
|
|
Substituting into the latter yields $\{u, v\} = \{u\}$.
|
|
Thus $u = v$ which in turn implies $v = y$.
|
|
|
|
\paragraph{Case 4}%
|
|
Suppose $\{u\} = \{x, y\}$ and $\{u, v\} = \{x, y\}$.
|
|
The former identity implies $x = y = u$.
|
|
Substituting into the latter yields $\{u, v\} = \{u\}$.
|
|
This implies $v = u$ which in turn implies $v = y$.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
These cases are exhaustive and each implies that $u = x$ and $v = y$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Lemma 3B}}%
|
|
\label{sub:lemma-3b}
|
|
|
|
\begin{theorem}[3B]
|
|
|
|
If $x \in C$ and $y \in C$, then
|
|
$\left< x, y \right> \in \powerset{\powerset{C}}$.
|
|
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_3}
|
|
{Enderton.Set.Chapter\_3.theorem\_3b}
|
|
|
|
Let $C$ be an arbitrary set and $x, y \in C$.
|
|
Then, by definition of the \nameref{ref:power-set},
|
|
$\{x\}$ and $\{x, y\}$ are members of $\powerset{C}$.
|
|
Likewise, $\{\{x\}, \{x, y\}\}$ is a member of $\powerset{\powerset{C}}$.
|
|
By definition of an \nameref{ref:ordered-pair},
|
|
$\left< x, y \right> = \{\{x\}, \{x, y\}\}$.
|
|
This concludes our proof.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Cartesian Product}}%
|
|
\label{sub:corollary-3c}
|
|
\label{sub:cartesian-product}
|
|
|
|
\begin{theorem}[3C]
|
|
|
|
For any sets $A$ and $B$, there is a set whose members are exactly the
|
|
pairs $\left< x, y \right>$ with $x \in A$ and $y \in B$.
|
|
|
|
\end{theorem}
|
|
|
|
\begin{proof}
|
|
|
|
\lean*{Mathlib/SetTheory/ZFC/Basic}{Set.prod}
|
|
|
|
\note{The above Lean proof is a definition (i.e. an axiom). It does not prove
|
|
such a set's existence from first principles.}
|
|
|
|
Define $C = A \cup B$.
|
|
Then for all $x \in A$ and for all $y \in B$, $x$ and $y$ are both in $C$.
|
|
By \nameref{sub:lemma-3b}, it follows that
|
|
$\left< x, y \right> \in \powerset{\powerset{C}}$.
|
|
The \nameref{ref:power-set-axiom} indicates $\powerset{\powerset{C}}$ is
|
|
indeed a set.
|
|
Therefore the \nameref{ref:subset-axioms} are applicable.
|
|
This implies the existence of a set $D$ such that
|
|
$$\forall z, (z \in D \iff z \in \powerset{\powerset{C}} \land
|
|
(\exists x, \exists y, x \in A \land y \in B \land
|
|
z = \left< x, y \right>)).$$
|
|
By construction $D$ is the set whose members are exactly the pairs
|
|
$\left< x, y \right>$ with $x \in A$ and $y \in B$.
|
|
|
|
\end{proof}
|
|
|
|
\section{Exercises 5}%
|
|
\label{sec:exercises-5}
|
|
|
|
\subsection{\verified{Exercise 5.1}}%
|
|
\label{sub:exercise-5.1}
|
|
|
|
Suppose that we attempted to generalize the Kuratowski definitions of ordered
|
|
pairs to ordered triples by defining
|
|
$$\left< x, y, z \right>^* = \{\{x\}, \{x, y\}, \{x, y, z\}\}.$$
|
|
Show that this definition is unsuccessful by giving examples of objects
|
|
$u$, $v$, $w$, $x$, $y$, $z$ with
|
|
$\left< x, y, z \right>^* = \left< u, v, w \right>^*$ but with either
|
|
$y \neq v$ or $z \neq w$ (or both).
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_3}
|
|
{Enderton.Set.Chapter\_3.exercise\_5\_1}
|
|
|
|
Let $x = 1$, $y = 1$, and $z = 2$.
|
|
Let $u = 1$, $v = 2$, and $w = 2$.
|
|
Then
|
|
\begin{align*}
|
|
\left< x, y, z \right>^*
|
|
& = \{\{x\}, \{x, y\}, \{x, y, z\}\} \\
|
|
& = \{\{1\}, \{1, 1\}, \{1, 1, 2\}\} \\
|
|
& = \{\{1\}, \{1, 2\}\}.
|
|
\end{align*}
|
|
Likewise
|
|
\begin{align*}
|
|
\left< u, v, w \right>^*
|
|
& = \{\{u\}, \{u, v\}, \{u, v, w\}\} \\
|
|
& = \{\{1\}, \{1, 2\}, \{1, 2, 2\}\} \\
|
|
& = \{\{1\}, \{1, 2\}\}.
|
|
\end{align*}
|
|
Thus $\left< x, y, z \right>^* = \left< u, v, w \right>^*$ but $y \neq v$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 5.2a}}%
|
|
\label{sub:exercise-5.2a}
|
|
|
|
Show that $A \times (B \cup C) = (A \times B) \cup (A \times C)$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_3}
|
|
{Enderton.Set.Chapter\_3.exercise\_5\_2a}
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets.
|
|
Then by definition of the \nameref{sub:cartesian-product} and union of sets,
|
|
\begin{align*}
|
|
A \times (B \cup C)
|
|
& = \{ \left< x, y \right> \mid x \in A \land y \in (B \cup C) \} \\
|
|
& = \{ \left< x, y \right> \mid
|
|
x \in A \land (y \in B \lor y \in C) \} \\
|
|
& = \{ \left< x, y \right> \mid
|
|
(x \in A \land y \in B) \lor (x \in A \land y \in C) \} \\
|
|
& = \{ \left< x, y \right> \mid (x \in A \land y \in B) \} \cup
|
|
\{ \left< x, y \right> \mid (x \in A \land y \in C) \} \\
|
|
& = (A \times B) \cup (A \times C).
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 5.2b}}%
|
|
\label{sub:exercise-5.2b}
|
|
|
|
Show that if $A \times B = A \times C$ and $A \neq \emptyset$, then $B = C$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_3}
|
|
{Enderton.Set.Chapter\_3.exercise\_5\_2b}
|
|
|
|
Let $A$, $B$, and $C$ be arbitrary sets such that $A \neq \emptyset$.
|
|
By definition of the \nameref{sub:cartesian-product},
|
|
\begin{align}
|
|
A \times B & = \{ \left< x, y \right> \mid x \in A \land y \in B \}
|
|
& \label{sub:exercise-5.2b-eq1} \\
|
|
A \times C & = \{ \left< x, y \right> \mid x \in A \land y \in C \}.
|
|
& \label{sub:exercise-5.2b-eq2}
|
|
\end{align}
|
|
There are two cases to consider:
|
|
|
|
\paragraph{Case 1}%
|
|
|
|
Suppose $B \neq \emptyset$.
|
|
Then $A \times B \neq \emptyset$ and $A \times C \neq \emptyset$.
|
|
Let $\left< x, y \right> \in A \times B$.
|
|
By \eqref{sub:exercise-5.2b-eq1}, $x \in A$ and $y \in B$.
|
|
By the \nameref{ref:extensionality-axiom},
|
|
$$\left< x, y \right> \in A \times B \iff \left< x, y \right> \in A \times C.$$
|
|
Therefore $\left< x, y \right> \in A \times C$.
|
|
By \eqref{sub:exercise-5.2b-eq2}, $x \in A$ and $y \in C$.
|
|
Since membership of $y$ in $B$ and in $C$ holds biconditionally, the
|
|
\nameref{ref:extensionality-axiom} indicates $B = C$.
|
|
|
|
\paragraph{Case 2}%
|
|
|
|
Suppose $B = \emptyset$.
|
|
Then there is no $\left< x, y \right>$ such that $x \in A$ and $y \in B$.
|
|
Thus $A \times B = \emptyset$ and $A \times C = \emptyset$.
|
|
But then there cannot exist an $\left< x, y \right>$ such that $x \in A$
|
|
and $y \in C$ either.
|
|
Since $A \neq \emptyset$, it must be the case that $C = \emptyset$.
|
|
Thus $B = C$.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\verified{Exercise 5.3}}%
|
|
\label{sub:exercise-5.3}
|
|
|
|
Show that $A \times \bigcup \mathscr{B} =
|
|
\bigcup\;\{ A \times X \mid X \in \mathscr{B} \}$.
|
|
|
|
\begin{proof}
|
|
|
|
\lean{Bookshelf/Enderton/Set/Chapter\_3}
|
|
{Enderton.Set.Chapter\_3.exercise\_5\_3}
|
|
|
|
Let $A$ and $\mathscr{B}$ be arbitrary sets.
|
|
By definition of the \nameref{sub:cartesian-product} and the union of sets,
|
|
\begin{align*}
|
|
A \times \bigcup\mathscr{B}
|
|
& = \{ \left< x, y \right> \mid
|
|
x \in A \land y \in \bigcup\mathscr{B} \} \\
|
|
& = \{ \left< x, y \right> \mid
|
|
x \in A \land (\exists b \in \mathscr{B}), y \in b \} \\
|
|
& = \{ \left< x, y \right> \mid
|
|
(\exists b \in \mathscr{B}), x \in A \land y \in b \} \\
|
|
& = \bigcup\; \{ A \times X \mid X \in \mathscr{B} \}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 5.4}}%
|
|
\label{sub:exercise-5.4}
|
|
|
|
Show that there is no set to which every ordered pair belongs.
|
|
|
|
\begin{proof}
|
|
|
|
For the sake of contradiction, suppose there exists a set $A$ to which every
|
|
ordered pair belongs.
|
|
That is, for all sets $x$ and $y$, $\left< x, y \right> = \{\{x\}, \{x, y\}\}$
|
|
is a member of $A$.
|
|
By the \nameref{ref:union-axiom}, it follows that $\bigcup\bigcup A$ is the
|
|
set to which every set belongs.
|
|
But \nameref{sub:theorem-2a} shows this is impossible.
|
|
Thus our original assumption was wrong; there exists no set to which every
|
|
ordered pair belongs.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 5.5a}}%
|
|
\label{sub:exercise-5.5a}
|
|
|
|
Assume that $A$ and $B$ are given sets, and show that there exists a set $C$
|
|
such that for any $y$,
|
|
$$y \in C \iff y = \{x\} \times B \text{ for some } x \text{ in } A.$$
|
|
In other words, show that $\{\{x\} \times B \mid x \in A\}$ is a set.
|
|
|
|
\begin{proof}
|
|
|
|
Let $A$ and $B$ be arbitrary sets.
|
|
Also let $x \in A$.
|
|
By definition of the \nameref{sub:cartesian-product},
|
|
\begin{equation}
|
|
\label{sub:exercise-5.5a-eq1}
|
|
\{x\} \times B = \{ \left< x, b \right> \mid b \in B \}.
|
|
\end{equation}
|
|
If $B = \emptyset$ then $\{x\} \times B$ trivially evaluates to the empty set,
|
|
which is a set by virtue of the \nameref{ref:empty-set-axiom}.
|
|
Therefore we continue under the assumption $B \neq \emptyset$.
|
|
|
|
We prove that (i)
|
|
$\{x\} \times B \subseteq \powerset{\powerset{(\{x\} \cup B)}}$ and then
|
|
(ii) that $\{x\} \times B$ is a set.
|
|
|
|
\paragraph{(i)}%
|
|
\label{par:exercise-5.5a-i}
|
|
|
|
Let $t \in \{x\} \times B$.
|
|
By \eqref{sub:exercise-5.5a-eq1} and definition of an
|
|
\nameref{ref:ordered-pair}, there exists a $b \in B$ such that
|
|
$$t = \left< x, b \right> = \{\{x\}, \{x, b\}\}.$$
|
|
It trivially holds that
|
|
$$\{x\} \subseteq \{x\} \cup B \quad\text{and}\quad
|
|
\{x, b\} \subseteq \{x\} \cup B.$$
|
|
Therefore, by definition of the \nameref{ref:power-set},
|
|
$$\{x\} \in \powerset{(\{x\} \cup B)} \quad\text{and}\quad
|
|
\{x, b\} \in \powerset{(\{x\} \cup B)}.$$
|
|
But then $\{\{x\}, \{x, b\}\} \subseteq \powerset{(\{x\} \cup B)}$.
|
|
Another application of the definition of the \nameref{ref:power-set} implies
|
|
that $$\{\{x\}, \{x, b\}\} \in \powerset{\powerset{(\{x\} \cup B)}}.$$
|
|
Since this holds for all sets $t$,
|
|
$\{x\} \times B \subseteq \powerset{\powerset{(\{x\} \cup B)}}$.
|
|
|
|
\paragraph{(ii)}%
|
|
\label{par:exercise-5.5a-ii}
|
|
|
|
By the \nameref{ref:subset-axioms}, there exists a set $C$ such that for any
|
|
set $y$, $$y \in C \iff
|
|
y \in \powerset{\powerset{(\{x\} \cup B)}} \land
|
|
(\exists b \in B, y = \left< x, b \right>).$$
|
|
The above equation and \eqref{sub:exercise-5.5a-eq1} implies $C$ contains
|
|
only ordered pairs of the desired form.
|
|
By \nameref{par:exercise-5.5a-i}, $C$ contains them all.
|
|
|
|
\paragraph{Conclusion}%
|
|
|
|
Since $x$ was an arbitrarily chosen member of $A$,
|
|
\nameref{par:exercise-5.5a-ii} immediately implies that
|
|
$\{\{x\} \times B \mid x \in A\}$ is indeed a set.
|
|
|
|
\end{proof}
|
|
|
|
\subsection{\partial{Exercise 5.5b}}%
|
|
\label{sub:exercise-5.5b}
|
|
|
|
With $A$, $B$, and $C$ as above, show that $A \times B = \bigcup C$.
|
|
|
|
\begin{proof}
|
|
|
|
Let $A$ and $B$ be arbitrary sets.
|
|
We want to show that
|
|
\begin{equation}
|
|
\label{sub:exercise-5.5b-eq1}
|
|
A \times B = \bigcup\; \{\{x\} \times B \mid x \in A\}.
|
|
\end{equation}
|
|
Note that \nameref{sub:cartesian-product} and \nameref{sub:exercise-5.5a}
|
|
prove the left- and right-hand sides of \eqref{sub:exercise-5.5b-eq1} are
|
|
sets respectively.
|
|
Then
|
|
\begin{align*}
|
|
A \times B
|
|
& = \{ y \mid \exists x \in A, \exists b \in B, y = \left< x, b \right> \} \\
|
|
& = \{ y \mid \exists b \in B, \exists x \in A, y = \left< x, b \right> \} \\
|
|
& = \{ y \mid \exists b \in B, y \in \{ \left< x, b \right> \mid x \in A \} \} \\
|
|
& = \{ y \mid y \in \{ \left< x, b \right> \mid x \in A \land b \in B \} \} \\
|
|
& = \{ y \mid \exists z \in \{\{x\} \times B \mid x \in A \}, y \in z \} \\
|
|
& = \bigcup \{\{x\} \times B \mid x \in A \}.
|
|
\end{align*}
|
|
|
|
\end{proof}
|
|
|
|
\end{document}
|