Exercise 9.1.1

Question

% Custom definitions (IntroToSetTheory)
% Source section: sec_9_1.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}}

% Section-local definitions
% Macros useful for this section
\def\uni{\bigcup_{i \in I}}
\def\sumi{\sum_{i \in I}}
\def\prodi{\prod_{i \in I}}
\def\prodj{\prod_{j \in J}}

ewcommand\unaj[1]{\bigcup_{j \in J_{#1}} A_j}
If $J_i$ ($i \in I$) are mutually disjoint sets and  $J = \bigcup_{i \in I} J_i$, and if $\k_j$ ($j \in J$) are cardinals, then
  \gath{
    \sum_{i \in I} \parens{ \sum_{j \in J_i} \k_j } = \sum_{j \in J} \k_j
  }
  (\emph{associativity} of $\sum$)

Dan Whitman · Accepted Answer

% Custom definitions (IntroToSetTheory)
% Source section: sec_9_1.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}}

% Section-local definitions
% Macros useful for this section
\def\uni{\bigcup_{i \in I}}
\def\sumi{\sum_{i \in I}}
\def\prodi{\prod_{i \in I}}
\def\prodj{\prod_{j \in J}}

ewcommand\unaj[1]{\bigcup_{j \in J_{#1}} A_j}
\qproof{
    Suppose that $\angles{A_j \where j \in J}$ are mutually disjoint sets where $\abs{A_j} = \k_j$ for every $j \in J$.
    Then, by definition, we have that
    \begin{gather}
      \sum_{j \in J} \k_j = \abs{\unaj{}}. \label{eqn:infsp:sumkjall}
    \end{gather}
    Now let $S = \braces{J_i \where i \in I}$, from which it is trivial to show that $\bigcup S = J$.
    It follows from Exercise~2.3.10 that
    \begin{gather}
      \unaj{} = \bigcup_{a \in \bigcup S} A_a = \bigcup_{C \in S} \parens{\bigcup_{a \in C} A_a}
      = \bigcup_{i \in I} \parens{\unaj{i}}. \label{eqn:infsp:dun}
    \end{gather}
    We claim that the sets $\angles{\unaj{i} \where i \in I}$ are mutually disjoint.
    So consider any $i_1$ and $i_2$ in $I$ where $i_1 
eq i_2$, and suppose that $\unaj{i_1}$ and $\unaj{i_2}$ are \emph{not} disjoint so that there is an $x$ where $x \in \unaj{i_1}$ and $x \in \unaj{i_2}$.
    Then there is a $j_1 \in J_{i_1}$ where $x \in A_{j_1}$ and a $j_2 \in J_{i_2}$ where $x \in A_{j_2}$.
    Now, since $\angles{J_i \where i \in I}$ are mutually disjoint and $i_1 
eq i_2$, it follows that $J_{i_1}$ and $J_{i_2}$ are disjoint.
    Therefore it has to be that $j_1 
eq j_2$ (since $j_1 \in J_{i_1}$ and $j_2 \in J_{i_2}$).
    But then $A_{j_1}$ and $A_{j_2}$ are not disjoint (since $x$ is in both) despite the fact that $j_1 
eq j_2$, which contradicts the fact that $\angles{A_j \where j \in J}$ are mutually disjoint.
    So it must be that in that $\unaj{i_1}$ and $\unaj{i_2}$ are disjoint, which proves the result since $i_1$ and $i_2$ were arbitrary.

Since $\angles{\unaj{i} \where i \in I}$ have been shown to be mutually disjoint, it follows by definition that
    \begin{gather}
      \sum_{i \in I} \abs{\unaj{i}} = \abs{\bigcup_{i \in I} \parens{\unaj{i}}}. \label{eqn:infsp:sumunaj}
    \end{gather}
    Lastly, we also clearly have that $\angles{A_j \where j \in J_i}$ are mutually disjoint for any $i \in I$ (since $\angles{A_j \where j \in J}$ are mutually disjoint) so that
    \begin{gather}
      \sum_{j \in J_i} \k_j = \abs{\unaj{i}}. \label{eqn:infsp:sumkj}
    \end{gather}

Putting this all together, we have
    \ali{
      \sum_{j \in J} \k_j &= \abs{\unaj{}} & 	ext{(by \eqref{eqn:infsp:sumkjall})} \
      &= \abs{\bigcup_{i \in I} \parens{\unaj{i}}} & 	ext{(by \eqref{eqn:infsp:dun})} \
      &= \sum_{i \in I} \abs{\unaj{i}} & 	ext{(by \eqref{eqn:infsp:sumunaj})} \
      &= \sum_{i \in I} \parens{\sum_{j \in J_i} \k_j} & 	ext{(by \eqref{eqn:infsp:sumkj})}
    }
    as desired.
  }

Exercise 9.1.1

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Exercise 9.1.1

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