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