Exercise 5.2.3

Question

% Custom definitions (IntroToSetTheory)
% Source section: sec_5_2.tex

% Common sets
\def
ats{{\boldsymbol{N}}}
\def\ints{{\boldsymbol{Z}}}
\defats{{\boldsymbol{Q}}}
\def\prats{ats^+}
\defeals{{\boldsymbol{R}}}
\def\es{\varnothing}

% Common variables/symbols
\def\vphi{\varphi}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\delta}
\def\e{\varepsilon}
\def\z{\zeta}
\def\k{\kappa}
\def\l{\lambda}
\def
{
u}
\def{ho}
\def\s{\sigma}
\def	{	au}
\def\x{\xi}
\def\w{\omega}
\def\W{\Omega}
\def\al{\aleph}

% Logic
\def\bic{\leftrightarrow}

ewcommand{\propP}[1]{\prop{P}(#1)}
\def\lxor{\veebar}

% For set-buider notation
\def\where{\mid}

% Other stuff
\def\dom{\mathrm{dom}\,}
\defan{\mathrm{ran}\,}

% Cardinality shortcuts
\def\cnats{{\al_0}}
\def\ccont{{2^\cnats}}

% Equivalence classes

ewcommand\eclass[2]{\squares{#1}_{#2}}

% Shortcuts that make writing easier

ewcommand\parens[1]{\left( #1 ight)}

ewcommand\squares[1]{\left[ #1 ight]}

ewcommand\braces[1]{\left\{ #1 ight\}}

ewcommand\angles[1]{\left\langle #1 ightangle}

ewcommand\ceil[1]{\left\lceil #1 ightceil}

ewcommand\floor[1]{\left\lfloor #1 ightfloor}

ewcommand\abs[1]{\left| #1 ight|}

ewcommand\dabs[1]{\left\| #1 ight\|}

ewcommand\vect[1]{\mathrm{\mathbf{#1}}}

ewcommand\conj[1]{\overline{#1}}

ewcommand\pset[1]{\mathcal{P}\left(#1ight)}

ewcommand\inv[1]{#1^{-1}}

ewcommand\prop[1]{\mathbf{#1}}
\defest{estriction}

ewcommand	et[2]{{^{#1}#2}}

% Families of set
\def\famF{\mathcal{F}}

% These are needed so that half-open intervals do not cause auto-indentation issues due to unmatched brackets

ewcommand\clop[1]{[#1)}

ewcommand\ilab[1]{#1)}

% Other miscellaneous stuff
\def\ss{\subseteq}
\def\pss{\subset}
\def\Seq{\mathrm{Seq}}
\def\prece{\preccurlyeq}
\def\sd{\bigtriangleup}

% Environment shortcuts

ewcommand\gath[1]{\begin{gather*} #1 \end{gather*}}

ewcommand\ali[1]{\begin{align*} #1 \end{align*}}

ewcommand\qproof[1]{\begin{proof} #1 \end{proof}}

ewcommand\bicproof[2]{
  $(	o)$ #1

$(\leftarrow)$ #2
}

ewcommand\seteqproof[2]{
  $(\ss)$ #1

$(\supseteq)$ #2
}

% Environment for indenting nested paragraphs (useful in case trees)

ewenvironment{indpar}
{
  \begin{adjustwidth}{1cm}{}
    }{
  \end{adjustwidth}
}

% Exercise, Theorem, and solution shortcuts

ewcommand\exercise[2]{{
      enewcommand\label[1]{} % Needed to suppress multiply-defined label warning since the same question numbers are used in different sections
      \setcounter{subsubsection}{#1-1} % Subsubsections are used to that lemmas have the full nested numbering
      \stepcounter{subsubsection} % Need to increment this so that the lemma counter gets reset
      \setcounter{question}{#1-1} % Manually set the question number since we always know the exercise number
      \question{#2} % The actual exam class question
    }}

ewcommand\exerciseapp[3]{
  \setqf{#2}
  \exercise{#1}{#3}
  \setqf{}
}

ewcommand	heorem[2]{{
      enewcommand\label[1]{} % Needed to suppress multiply-defined label warning since the same question numbers are used in different sections
      \setcounter{subsubsection}{#1-1} % Subsubsections are used to that lemmas have the full nested numbering
      \stepcounter{subsubsection} % Need to increment this so that the lemma counter gets reset
      \setcounter{question}{#1-1} % Manually set the question number since we always know the theorem number
      \question{#2} % The actual exam class question
    }}

ewcommand	heoremapp[3]{
  \setqf{#2}
  	heorem{#1}{#3}
  \setqf{}
}

ewcommand\sol[1]{
  \begin{solution}

#1
  \end{solution}
}

% Main problem and theorem labels
\def\mainprob{	extbf{Main Problem.}}
\def\mainthrm{	extbf{Main Theorem.}}

% Saves some typing
\def\cbthrm{Cantor-Bernstein Theorem}

% Book title
\def\booktitle{
  Introduction to Set Theory \
  Third Edition, Revised and Expanded \
  by Karel Hrbacek and Thomas Jech
}

% Environment aliases used in the source sections

ewenvironment{lem}{\begin{lemma}}{\end{lemma}}

ewenvironment{defin}{\begin{definition}}{\end{definition}}

ewenvironment{cor}{\begin{corollary}}{\end{corollary}}

ewenvironment{thrm}{\begin{theorem}}{\end{theorem}}

% Override indpar to avoid changepage/adjustwidth dependency
enewenvironment{indpar}{\begin{quote}}{\end{quote}}
If a linearly ordered set $P$ has a countable dense subset, then $\abs{P} \leq 2^{\al_0}$.

Dan Whitman · Accepted Answer

% Custom definitions (IntroToSetTheory)
% Source section: sec_5_2.tex

% Common sets
\def
ats{{\boldsymbol{N}}}
\def\ints{{\boldsymbol{Z}}}
\defats{{\boldsymbol{Q}}}
\def\prats{ats^+}
\defeals{{\boldsymbol{R}}}
\def\es{\varnothing}

% Common variables/symbols
\def\vphi{\varphi}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\delta}
\def\e{\varepsilon}
\def\z{\zeta}
\def\k{\kappa}
\def\l{\lambda}
\def
{
u}
\def{ho}
\def\s{\sigma}
\def	{	au}
\def\x{\xi}
\def\w{\omega}
\def\W{\Omega}
\def\al{\aleph}

% Logic
\def\bic{\leftrightarrow}

ewcommand{\propP}[1]{\prop{P}(#1)}
\def\lxor{\veebar}

% For set-buider notation
\def\where{\mid}

% Other stuff
\def\dom{\mathrm{dom}\,}
\defan{\mathrm{ran}\,}

% Cardinality shortcuts
\def\cnats{{\al_0}}
\def\ccont{{2^\cnats}}

% Equivalence classes

ewcommand\eclass[2]{\squares{#1}_{#2}}

% Shortcuts that make writing easier

ewcommand\parens[1]{\left( #1 ight)}

ewcommand\squares[1]{\left[ #1 ight]}

ewcommand\braces[1]{\left\{ #1 ight\}}

ewcommand\angles[1]{\left\langle #1 ightangle}

ewcommand\ceil[1]{\left\lceil #1 ightceil}

ewcommand\floor[1]{\left\lfloor #1 ightfloor}

ewcommand\abs[1]{\left| #1 ight|}

ewcommand\dabs[1]{\left\| #1 ight\|}

ewcommand\vect[1]{\mathrm{\mathbf{#1}}}

ewcommand\conj[1]{\overline{#1}}

ewcommand\pset[1]{\mathcal{P}\left(#1ight)}

ewcommand\inv[1]{#1^{-1}}

ewcommand\prop[1]{\mathbf{#1}}
\defest{estriction}

ewcommand	et[2]{{^{#1}#2}}

% Families of set
\def\famF{\mathcal{F}}

% These are needed so that half-open intervals do not cause auto-indentation issues due to unmatched brackets

ewcommand\clop[1]{[#1)}

ewcommand\ilab[1]{#1)}

% Other miscellaneous stuff
\def\ss{\subseteq}
\def\pss{\subset}
\def\Seq{\mathrm{Seq}}
\def\prece{\preccurlyeq}
\def\sd{\bigtriangleup}

% Environment shortcuts

ewcommand\gath[1]{\begin{gather*} #1 \end{gather*}}

ewcommand\ali[1]{\begin{align*} #1 \end{align*}}

ewcommand\qproof[1]{\begin{proof} #1 \end{proof}}

ewcommand\bicproof[2]{
  $(	o)$ #1

$(\leftarrow)$ #2
}

ewcommand\seteqproof[2]{
  $(\ss)$ #1

$(\supseteq)$ #2
}

% Environment for indenting nested paragraphs (useful in case trees)

ewenvironment{indpar}
{
  \begin{adjustwidth}{1cm}{}
    }{
  \end{adjustwidth}
}

% Exercise, Theorem, and solution shortcuts

ewcommand\exercise[2]{{
      enewcommand\label[1]{} % Needed to suppress multiply-defined label warning since the same question numbers are used in different sections
      \setcounter{subsubsection}{#1-1} % Subsubsections are used to that lemmas have the full nested numbering
      \stepcounter{subsubsection} % Need to increment this so that the lemma counter gets reset
      \setcounter{question}{#1-1} % Manually set the question number since we always know the exercise number
      \question{#2} % The actual exam class question
    }}

ewcommand\exerciseapp[3]{
  \setqf{#2}
  \exercise{#1}{#3}
  \setqf{}
}

ewcommand	heorem[2]{{
      enewcommand\label[1]{} % Needed to suppress multiply-defined label warning since the same question numbers are used in different sections
      \setcounter{subsubsection}{#1-1} % Subsubsections are used to that lemmas have the full nested numbering
      \stepcounter{subsubsection} % Need to increment this so that the lemma counter gets reset
      \setcounter{question}{#1-1} % Manually set the question number since we always know the theorem number
      \question{#2} % The actual exam class question
    }}

ewcommand	heoremapp[3]{
  \setqf{#2}
  	heorem{#1}{#3}
  \setqf{}
}

ewcommand\sol[1]{
  \begin{solution}

#1
  \end{solution}
}

% Main problem and theorem labels
\def\mainprob{	extbf{Main Problem.}}
\def\mainthrm{	extbf{Main Theorem.}}

% Saves some typing
\def\cbthrm{Cantor-Bernstein Theorem}

% Book title
\def\booktitle{
  Introduction to Set Theory \
  Third Edition, Revised and Expanded \
  by Karel Hrbacek and Thomas Jech
}

% Environment aliases used in the source sections

ewenvironment{lem}{\begin{lemma}}{\end{lemma}}

ewenvironment{defin}{\begin{definition}}{\end{definition}}

ewenvironment{cor}{\begin{corollary}}{\end{corollary}}

ewenvironment{thrm}{\begin{theorem}}{\end{theorem}}

% Override indpar to avoid changepage/adjustwidth dependency
enewenvironment{indpar}{\begin{quote}}{\end{quote}}
Note that the countable dense subset is dense in $P$ and not just in itself as explained in the errata list.
  \qproof{
    Suppose that $(P,<)$ is our linearly ordered set and $R$ is the countable dense subset of $P$.
    We construct an $f : P 	o \pset{R}$ by defining
    $$
      f(x) = \braces{y \in R \where y < x}
    $$
    for any $x \in P$.
    Clearly for such $x$ we have that $f(x) \ss R$ so that $f(x) \in \pset{R}$.

Now we claim that $f$ is injective.
    So consider any $x,y \in P$ such that $x 
eq y$.
    Without loss of generality we can assume that $x < y$.
    Since $R$ is dense in $P$ there is a $z \in R$ where $x < z < y$.
    From this it follows that $z \in f(y)$ but that $z 
otin f(x)$ since it is not true that $z < x$.
    Hence clearly $f(x) 
eq f(y)$ so that we have shown that $f$ is injective.

Thus we have
    $$
      |P| \leq |\pset{R}| = 2^{|R|} = \ccont,
    $$
    where we have  used Theorem~5.1.9.
  }

Exercise 5.2.3

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Exercise 5.2.3

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