Enderton (set). Add cardinal arithmetic theorems and exercise prompts.

Bookshelf/Enderton/Set.tex

@@ -77,6 +77,19 @@
 
   \lean{Mathlib/Data/Set/Prod}{Set.prod}
 
+\section{\defined{Cardinal Arithmetic}}%
+\hyperlabel{sec:cardinal-arithmetic}
+
+  Let $\kappa$ and $\lambda$ be any cardinal numbers.
+  \begin{enumerate}[(a)]
+    \item $\kappa + \lambda = \card{(K \cup L)}$, where $K$ and $L$ are any
+      disjoint sets of cardinality $\kappa$ and $\lambda$, respectively.
+    \item $\kappa \cdot \lambda = \card{(K \times L)}$, where $K$ and $L$ are
+      any sets of cardinality $\kappa$ and $\lambda$, respectively.
+    \item $\kappa^\lambda = \card{^L{K}}$, where $K$ and $L$ are any sets of
+      cardinality $\kappa$ and $\lambda$, respectively.
+  \end{enumerate}
+
 \section{\defined{Compatible}}%
 \hyperlabel{ref:compatible}
 
@@ -8811,11 +8824,73 @@
     Hence $S'$ is a finite set.
   \end{proof}
 
+\subsection{\sorry{Theorem 6H}}%
+\hyperlabel{sub:theorem-6h}
+
+  Assume that $\equinumerous{K_1}{K_2}$ and $\equinumerous{L_1}{L_2}$.
+    \begin{enumerate}[(a)]
+      \item If $K_1 \cap L_1 = K_2 \cap L_2 = \emptyset$, then
+        $\equinumerous{K_1 \cup L_1}{K_2 \cup L_2}$.
+      \item $\equinumerous{K_1 \times L_1}{K_2 \times L_2}$.
+      \item $\equinumerous{^{(L_1)}{K_1}}{^{(L_2)}{K_2}}$.
+    \end{enumerate}
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Theorem 6I}}%
+\hyperlabel{sub:theorem-6i}
+
+  For any cardinal numbers $\kappa$, $\lambda$, and $\mu$:
+    \begin{enumerate}
+      \item $\kappa + \lambda = \lambda + \kappa$ and
+        $\kappa \cdot \lambda = \lambda \cdot \kappa$.
+      \item $\kappa + (\lambda + \mu) = (\kappa + \lambda) + \mu$ and
+        $\kappa \cdot (\lambda \cdot \mu) = (\kappa \cdot \lambda) \cdot \mu$.
+      \item $\kappa \cdot (\lambda + \mu) =
+        \kappa \cdot \lambda + \kappa \cdot \mu$.
+      \item $\kappa^{\lambda + \mu} = \kappa^\lambda \cdot \kappa^\mu$.
+      \item $(\kappa \cdot \lambda)^\mu = \kappa^\mu \cdot \lambda^\mu$.
+      \item $(\kappa^\lambda)^\mu = \kappa^{\lambda \cdot \mu}$.
+    \end{enumerate}
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Theorem 6J}}%
+\hyperlabel{sub:theorem-6j}
+
+  Let $m$ and $n$ be finite cardinals.
+  Then
+    \begin{align*}
+      m + n & = m +_\omega n, \\
+      m \cdot n = m \cdot_\omega n, \\
+      m^n = m^n,
+    \end{align*}
+    where on the right side we use the operations of $\omega$ defined via
+    recursion and on the left side we use the operations of cardinal arithmetic.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Corollary 6K}}%
+\hyperlabel{sub:corollary-6k}
+
+  If $A$ and $B$ are finite, then $A \cup B$, $A \times B$, and $^B{A}$ are also
+    finite.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
 \section{Exercises 6}%
 \hyperlabel{sec:exercises-6}
 
 \subsection{\unverified{Exercise 6.1}}%
-\hyperlabel{sub:exercise-6-1}
+\hyperlabel{sub:exercise-6.1}
 
   Show that the equation $$f(m, n) = 2^m(2n + 1) - 1$$ defines a one-one-one
     correspondence between $\omega \times \omega$ and $\omega$.
@@ -8894,7 +8969,7 @@
   \end{proof}
 
 \subsection{\unverified{Exercise 6.2}}%
-\hyperlabel{sub:exercise-6-2}
+\hyperlabel{sub:exercise-6.2}
 
   Show that in Fig. 32 we have:
     \begin{align*}
@@ -8933,7 +9008,7 @@
   \end{proof}
 
 \subsection{\unverified{Exercise 6.3}}%
-\hyperlabel{sub:exercise-6-3}
+\hyperlabel{sub:exercise-6.3}
 
   Find a one-to-one correspondence between the open unit interval $\ioo{0}{1}$
     and $\mathbb{R}$ that takes rationals to rationals and irrationals to
@@ -9087,7 +9162,7 @@
   \end{proof}
 
 \subsection{\sorry{Exercise 6.4}}%
-\hyperlabel{sub:exercise-6-4}
+\hyperlabel{sub:exercise-6.4}
 
   Construct a one-to-one correspondence between the closed unit interval
     $$\icc{0}{1} = \{x \in \mathbb{R} \mid 0 \leq x \leq 1\}$$
@@ -9098,7 +9173,7 @@
   \end{proof}
 
 \subsection{\verified{Exercise 6.5}}%
-\hyperlabel{sub:exercise-6-5}
+\hyperlabel{sub:exercise-6.5}
 
   Prove \nameref{sub:theorem-6a}.
 
@@ -9107,7 +9182,7 @@
   \end{proof}
 
 \subsection{\sorry{Exercise 6.6}}%
-\hyperlabel{sub:exercise-6-6}
+\hyperlabel{sub:exercise-6.6}
 
   Let $\kappa$ be a nonzero cardinal number.
   Show there does not exist a set to which every set of cardinality $\kappa$
@@ -9118,7 +9193,7 @@
   \end{proof}
 
 \subsection{\sorry{Exercise 6.7}}%
-\hyperlabel{sub:exercise-6-7}
+\hyperlabel{sub:exercise-6.7}
 
   Assume that $A$ is finite and $f \colon A \rightarrow A$.
   Show that $f$ is one-to-one iff $\ran{f} = A$.
@@ -9128,7 +9203,7 @@
   \end{proof}
 
 \subsection{\sorry{Exercise 6.8}}%
-\hyperlabel{sub:exercise-6-8}
+\hyperlabel{sub:exercise-6.8}
 
   Prove that the union of two finite sets is finite, without any use of
     arithmetic.
@@ -9138,7 +9213,7 @@
   \end{proof}
 
 \subsection{\sorry{Exercise 6.9}}%
-\hyperlabel{sub:exercise-6-9}
+\hyperlabel{sub:exercise-6.9}
 
   Prove that the Cartesian product of two finite sets is finite, without any use
     of arithmetic.
@@ -9147,4 +9222,60 @@
     TODO
   \end{proof}
 
+\subsection{\sorry{Exercise 6.10}}%
+\hyperlabel{sub:exercise-6.10}
+
+  Prove part 4 of \nameref{sub:theorem-6i}.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Exercise 6.11}}%
+\hyperlabel{sub:exercise-6.11}
+
+  Prove part 5 of \nameref{sub:theorem-6i}.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Exercise 6.12}}%
+\hyperlabel{sub:exercise-6.12}
+
+  The proof to \nameref{sub:theorem-6i} involves eight instances of showing two
+    sets to be equinumerous.
+  (The eight are listed in the proof of the theorem as statements numbered 1-6.)
+  In which of these eight cases does equality actually hold?
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Exercise 6.13}}%
+\hyperlabel{sub:exercise-6.13}
+
+  Show that a finite union of finite sets is finite.
+  That is, show that if $B$ is a finite set whose members are themselves finite
+    sets, then $\bigcup{B}$ is finite.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
+\subsection{\sorry{Exercise 6.14}}%
+\hyperlabel{sub:exercise-6.14}
+
+  Define a \textit{permutation} of $K$ to be any one-to-one function from $K$
+    onto $K$.
+  We can the define the factorial operation on cardinal numbers by the equation
+    $$\kappa! = \card{\{f \mid f \text{ is a permutation of } K\}},$$
+    where $K$ is any set of cardinality $\kappa$.
+  Show that $\kappa!$ is well defined, i.e. the value of $\kappa!$ is
+    independent of just which set $K$ is chosen.
+
+  \begin{proof}
+    TODO
+  \end{proof}
+
 \end{document}

preamble.tex

@@ -166,6 +166,7 @@
 % ========================================
 
 \newcommand{\abs}[1]{\left|#1\right|}
+\newcommand{\card}[1]{\mathop{\text{card}}{#1}}
 \newcommand{\ceil}[1]{\left\lceil#1\right\rceil}
 \newcommand{\ctuple}[2]{\left< #1, \cdots, #2 \right>}
 \newcommand{\dom}[1]{\textop{dom}{#1}}