Enderton. Function question prompts.

Bookshelf/Enderton/Set.tex

@@ -3644,4 +3644,289 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
 \end{proof}
 
+\section{Exercise 8}%
+\label{sec:exercise-8}
+
+\subsection{\unverified{Exercise 8.11}}%
+\label{sub:exercise-8.11}
+
+Prove the following version (for functions) of the extensionality principle:
+  Assume that $F$ and $G$ are functions, $\dom{F} = \dom{G}$, and
+  $F(x) = G(x)$ for all $x$ in the common domain.
+Then $F = G$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.12}}%
+\label{sub:exercise-8.12}
+
+Assume that $f$ and $g$ are functions and show that
+  $$f \subseteq g \iff \dom{f} \subseteq \dom{g} \land
+    (\forall x \in \dom{f}) f(x) = g(x).$$
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.13}}%
+\label{sub:exercise-8.13}
+
+Assume that $f$ and $g$ are functions with $f \subseteq g$ and
+  $\dom{g} \subseteq \dom{f}$.
+Show that $f = g$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.14}}%
+\label{sub:exercise-8.14}
+
+Assume that $f$ and $g$ are functions.
+
+\begin{enumerate}[(a)]
+  \item Show that $f \cap g$ is a function.
+  \item Show that $f \cup g$ is a function iff $f(x) = g(x)$ for every $x$ in
+    $(\dom{f}) \cap (\dom{g})$.
+\end{enumerate}
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.15}}%
+\label{sub:exercise-8.15}
+
+Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in
+  $\mathscr{A}$, either $f \subseteq g$ or $g \subseteq f$.
+Show that $\bigcup \mathscr{A}$ is a function.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.16}}%
+\label{sub:exercise-8.16}
+
+Show that there is no set to which every function belongs.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.17}}%
+\label{sub:exercise-8.17}
+
+Show that the composition of two single-rooted sets is again single-rooted.
+Conclude that the composition of two one-to-one functions is again one-to-one.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.18}}%
+\label{sub:exercise-8.18}
+
+Let $R$ be the set
+  $$\{ \left< 0, 1 \right>, \left< 0, 2 \right>, \left< 0, 3 \right>,
+       \left< 1, 2 \right>, \left< 1, 3 \right>, \left< 2, 3 \right>\}.$$
+Evaluate the following: $R \circ R$, $R \restriction \{1\}$,
+  $R^{-1} \restriction \{1\}$, $\img{R}{\{1\}}$, and $\img{R^{-1}}{\{1\}}$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.19}}%
+\label{sub:exercise-8.19}
+
+Let $$A = \{
+  \left< \emptyset, \{\emptyset, \{\emptyset\}\} \right>,
+  \left< \{\emptyset\}, \emptyset \right>
+  \}.$$
+Evaluate each of the following: $A(\emptyset)$, $\img{A}{\emptyset}$,
+  $\img{A}{\{\emptyset\}}$, $\img{A}{\{\emptyset, \{\emptyset\}\}}$,
+  $A^{-1}$, $A \circ A$, $A \restriction \emptyset$,
+  $A \restriction \{\emptyset\}$, $A \restriction \{\emptyset, \{\emptyset\}\}$,
+  $\bigcup\bigcup A$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.20}}%
+\label{sub:exercise-8.20}
+
+Show that $F \restriction A = F \cap (A \times \ran{F})$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.21}}%
+\label{sub:exercise-8.21}
+
+Show that $(R \circ S) \circ T = R \circ (S \circ T)$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.22}}%
+\label{sub:exercise-8.22}
+
+Show that the following are correct for any sets.
+
+\begin{enumerate}[(a)]
+  \item $A \subseteq B \Rightarrow \img{F}{A} \subseteq \img{F}{B}$.
+  \item $\img{(F \circ G)}{A} = \img{F}{\img{G}{A}}$.
+  \item $Q \restriction (A \cup B) =
+    (Q \restriction A) \cup (Q \restriction B)$.
+\end{enumerate}
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.23}}%
+\label{sub:exercise-8.23}
+
+Let $I_A$ be the identity function on the set $A$.
+Show that for any sets $B$ and $C$,
+  $$B \circ I_A = B \restriction A \quad\text{and}\quad
+    \img{I_A}{C} = A \cap C.$$
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.24}}%
+\label{sub:exercise-8.24}
+
+Show that for a function $F$,
+  $\img{F^{-1}}{A} = \{x \in \dom{F} \mid F(x) \in A\}$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.25}}%
+\label{sub:exercise-8.25}
+
+\begin{enumerate}[(a)]
+  \item Assume that $G$ is a one-to-one function.
+    Show that $G \circ G^{-1}$ is $I_{\ran{G}}$, the identity function on
+      $\ran{G}$.
+  \item Show that the result of part (a) holds for any function $G$, not
+    necessarily one-to-one.
+\end{enumerate}
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.26}}%
+\label{sub:exercise-8.26}
+
+Prove the second halves of parts (a) and (b) of Theorem 3K.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.27}}%
+\label{sub:exercise-8.27}
+
+Show that $\dom{(F \circ G)} = \img{G^{-1}}{\dom{F}}$ for any sets $F$ and $G$.
+($F$ and $G$ need not be functions.)
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.28}}%
+\label{sub:exercise-8.28}
+
+Assume that $f$ is a one-to-one function from $A$ into $B$, and that $G$ is the
+  function with $\dom{G} = \powerset{A}$ defined by the equation
+  $G(X) = \img{f}{x}$.
+Show that $G$ maps $\powerset{A}$ one-to-one into $\powerset{B}$.
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.29}}%
+\label{sub:exercise-8.29}
+
+Assume that $f \colon A \rightarrow B$ and define a function
+  $G \colon B \rightarrow \powerset{A}$ by
+  $$G(b) = \{x \in A \mid f(x) = b\}.$$
+Show that if $f$ maps $A$ \textit{onto} $B$, then $G$ is one-to-one.
+Does the converse hold?
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
+\subsection{\unverified{Exercise 8.30}}%
+\label{sub:exercise-8.30}
+
+Assume that $F \colon \powerset{A} \rightarrow \powerset{A}$ and that $F$ has
+  the monotonicity property:
+  $$X \subseteq Y \subseteq A \Rightarrow F(X) \subseteq F(Y).$$
+Define
+  $$B = \bigcap\{X \subseteq A \mid F(X) \subseteq X\} \quad\text{and}\quad
+    C = \bigcup\{X \subseteq A \mid X \subseteq F(X)\}.$$
+\begin{enumerate}[(a)]
+  \item Show that $F(B) = B$ and $F(C) = C$.
+  \item Show that if $F(X) = X$, then $B \subseteq X \subseteq C$.
+\end{enumerate}
+
+\begin{proof}
+
+  TODO
+
+\end{proof}
+
 \end{document}