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