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Update proofs to pending if there is no custom proof set.

finite-set-exercises
Joshua Potter 2023-08-08 17:30:29 -06:00
parent b91ee81261
commit ec53cdd710
1 changed files with 33 additions and 32 deletions
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Bookshelf/Enderton/Set.tex
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@ -1114,7 +1114,7 @@ List all the members of $V_4$.
\section{Algebra of Sets}% \section{Algebra of Sets}%
\hyperlabel{sec:algebra-sets} \hyperlabel{sec:algebra-sets}
\subsection{\verified{Commutative Laws}}% \subsection{\pending{Commutative Laws}}%
\hyperlabel{sub:commutative-laws} \hyperlabel{sub:commutative-laws}
For any sets $A$ and $B$, For any sets $A$ and $B$,
@ -1160,7 +1160,7 @@ For any sets $A$ and $B$,
\end{proof} \end{proof}
\subsection{\verified{Associative Laws}}% \subsection{\pending{Associative Laws}}%
\hyperlabel{sub:associative-laws} \hyperlabel{sub:associative-laws}
For any sets $A$, $B$ and $C$, For any sets $A$, $B$ and $C$,
@ -1217,7 +1217,7 @@ For any sets $A$, $B$ and $C$,
\end{proof} \end{proof}
\subsection{\verified{Distributive Laws}}% \subsection{\pending{Distributive Laws}}%
\hyperlabel{sub:distributive-laws} \hyperlabel{sub:distributive-laws}
For any sets $A$, $B$, and $C$, For any sets $A$, $B$, and $C$,
@ -1273,7 +1273,7 @@ For any sets $A$, $B$, and $C$,
\end{proof} \end{proof}
\subsection{\verified{De Morgan's Laws}}% \subsection{\pending{De Morgan's Laws}}%
\hyperlabel{sub:de-morgans-laws} \hyperlabel{sub:de-morgans-laws}
For any sets $A$, $B$, and $C$, For any sets $A$, $B$, and $C$,
@ -1331,7 +1331,7 @@ For any sets $A$, $B$, and $C$,
\end{proof} \end{proof}
\subsection{\verified{% \subsection{\pending{%
Identities Involving \texorpdfstring{$\emptyset$}{the Empty Set}}}% Identities Involving \texorpdfstring{$\emptyset$}{the Empty Set}}}%
\hyperlabel{sub:identitives-involving-empty-set} \hyperlabel{sub:identitives-involving-empty-set}
@ -1401,7 +1401,7 @@ For any set $A$,
\end{proof} \end{proof}
\subsection{\verified{Monotonicity}}% \subsection{\pending{Monotonicity}}%
\hyperlabel{sub:monotonicity} \hyperlabel{sub:monotonicity}
For any sets $A$, $B$, and $C$, For any sets $A$, $B$, and $C$,
@ -1474,7 +1474,7 @@ For any sets $A$, $B$, and $C$,
\end{proof} \end{proof}
\subsection{\verified{Anti-monotonicity}}% \subsection{\pending{Anti-monotonicity}}%
\hyperlabel{sub:anti-monotonicity} \hyperlabel{sub:anti-monotonicity}
For any sets $A$, $B$, and $C$, For any sets $A$, $B$, and $C$,
@ -1644,7 +1644,7 @@ For any set $C$ and $\mathscr{A} \neq \emptyset$,
\end{proof} \end{proof}
\subsection{\verified{% \subsection{\pending{%
\texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}% \texorpdfstring{$\cap$/$-$}{Intersection/Difference} Associativity}}%
\hyperlabel{sub:intersection-difference-associativity} \hyperlabel{sub:intersection-difference-associativity}
@ -2187,7 +2187,7 @@ Show by example that for some sets $A$, $B$, and $C$, the set $A - (B - C)$ is
\end{proof} \end{proof}
\subsection{\verified{Exercise 2.15a}}% \subsection{\pending{Exercise 2.15a}}%
\hyperlabel{sub:exercise-2.15a} \hyperlabel{sub:exercise-2.15a}
Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$. Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$.
@ -2224,7 +2224,7 @@ Show that $A \cap (B + C) = (A \cap B) + (A \cap C)$.
\end{proof} \end{proof}
\subsection{\verified{Exercise 2.15b}}% \subsection{\pending{Exercise 2.15b}}%
\hyperlabel{sub:exercise-2.15b} \hyperlabel{sub:exercise-2.15b}
Show that $A + (B + C) = (A + B) + C$. Show that $A + (B + C) = (A + B) + C$.
@ -2914,7 +2914,7 @@ If not, then under what conditions does equality hold?
\end{proof} \end{proof}
\subsection{\verified{Corollary 3C}}% \subsection{\pending{Corollary 3C}}%
\hyperlabel{sub:corollary-3c} \hyperlabel{sub:corollary-3c}
\begin{theorem}[3C] \begin{theorem}[3C]
@ -3730,7 +3730,8 @@ If not, then under what conditions does equality hold?
\begin{proof} \begin{proof}
\lean{Bookshelf/Enderton/Set/Relation}{Set.Relation.modEquiv\_partition} \lean{Bookshelf/Enderton/Set/Relation}
{Set.Relation.modEquiv\_partition}
Let $\Pi = \{[x]_R \mid x \in A\}$. Let $\Pi = \{[x]_R \mid x \in A\}$.
We show that (i) there are no empty sets in $\Pi$, (ii) no two different sets We show that (i) there are no empty sets in $\Pi$, (ii) no two different sets
@ -4345,7 +4346,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
\end{answer} \end{answer}
\subsection{\verified{Exercise 3.11}}% \subsection{\pending{Exercise 3.11}}%
\hyperlabel{sub:exercise-3.11} \hyperlabel{sub:exercise-3.11}
Prove the following version (for functions) of the extensionality principle: Prove the following version (for functions) of the extensionality principle:
@ -6527,7 +6528,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\section{Arithmetic}% \section{Arithmetic}%
\hyperlabel{sec:arithmetic} \hyperlabel{sec:arithmetic}
\subsection{\verified{Theorem 4I}} \subsection{\pending{Theorem 4I}}
\hyperlabel{sub:theorem-4i} \hyperlabel{sub:theorem-4i}
\begin{theorem}[4I] \begin{theorem}[4I]
@ -6563,7 +6564,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4J}} \subsection{\pending{Theorem 4J}}
\hyperlabel{sub:theorem-4j} \hyperlabel{sub:theorem-4j}
\begin{theorem}[4J] \begin{theorem}[4J]
@ -6597,7 +6598,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Left Additive Identity}}% \subsection{\pending{Left Additive Identity}}%
\hyperlabel{sub:left-additive-identity} \hyperlabel{sub:left-additive-identity}
\begin{lemma} \begin{lemma}
@ -6639,7 +6640,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Lemma 3}}% \subsection{\pending{Lemma 3}}%
\hyperlabel{sub:lemma-3} \hyperlabel{sub:lemma-3}
\hyperlabel{sub:succ-add-eq-add-succ} \hyperlabel{sub:succ-add-eq-add-succ}
@ -6683,7 +6684,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4K-1}}% \subsection{\pending{Theorem 4K-1}}%
\label{sub:theorem-4k-1} \label{sub:theorem-4k-1}
\begin{theorem}[4K-1] \begin{theorem}[4K-1]
@ -6739,7 +6740,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4K-2}}% \subsection{\pending{Theorem 4K-2}}%
\label{sub:theorem-4k-2} \label{sub:theorem-4k-2}
\begin{theorem}[4K-2] \begin{theorem}[4K-2]
@ -6791,7 +6792,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Zero Multiplicand}}% \subsection{\pending{Zero Multiplicand}}%
\hyperlabel{sub:zero-multiplicand} \hyperlabel{sub:zero-multiplicand}
\begin{lemma} \begin{lemma}
@ -6841,7 +6842,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Successor Distribution}}% \subsection{\pending{Successor Distribution}}%
\hyperlabel{sub:successor-distribution} \hyperlabel{sub:successor-distribution}
\begin{lemma} \begin{lemma}
@ -6897,7 +6898,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4K-3}} \subsection{\pending{Theorem 4K-3}}
\hyperlabel{sub:theorem-4k-3} \hyperlabel{sub:theorem-4k-3}
\begin{theorem}[4K-3] \begin{theorem}[4K-3]
@ -6958,7 +6959,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Successor Identity}}% \subsection{\pending{Successor Identity}}%
\hyperlabel{sub:successor-identity} \hyperlabel{sub:successor-identity}
\begin{lemma} \begin{lemma}
@ -7011,7 +7012,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Right Multiplicative Identity}}% \subsection{\pending{Right Multiplicative Identity}}%
\hyperlabel{sub:right-multiplicative-identity} \hyperlabel{sub:right-multiplicative-identity}
\begin{lemma} \begin{lemma}
@ -7062,7 +7063,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4K-5}} \subsection{\pending{Theorem 4K-5}}
\hyperlabel{sub:theorem-4k-5} \hyperlabel{sub:theorem-4k-5}
\begin{theorem}[4K-5] \begin{theorem}[4K-5]
@ -7122,7 +7123,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4K-4}}% \subsection{\pending{Theorem 4K-4}}%
\hyperlabel{sub:theorem-4k-4} \hyperlabel{sub:theorem-4k-4}
\begin{theorem}[4K-4] \begin{theorem}[4K-4]
@ -7180,7 +7181,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\section{Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}% \section{Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}%
\hyperlabel{sec:ordering-natural-numbers} \hyperlabel{sec:ordering-natural-numbers}
\subsection{\verified{Ordering on Successor}}% \subsection{\pending{Ordering on Successor}}%
\hyperlabel{sub:ordering-successor} \hyperlabel{sub:ordering-successor}
\begin{lemma} \begin{lemma}
@ -7226,7 +7227,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Lemma 4L(a)}}% \subsection{\pending{Lemma 4L(a)}}%
\hyperlabel{sub:lemma-4l-a} \hyperlabel{sub:lemma-4l-a}
\begin{lemma}[4L(a)] \begin{lemma}[4L(a)]
@ -7294,7 +7295,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Lemma 4L(b)}}% \subsection{\pending{Lemma 4L(b)}}%
\hyperlabel{sub:lemma-4l-b} \hyperlabel{sub:lemma-4l-b}
\begin{lemma}[4L(b)] \begin{lemma}[4L(b)]
@ -7577,7 +7578,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Theorem 4N}}% \subsection{\pending{Theorem 4N}}%
\hyperlabel{sub:theorem-4n} \hyperlabel{sub:theorem-4n}
\begin{theorem}[4N] \begin{theorem}[4N]
@ -7695,7 +7696,7 @@ Show that $<_L$ is a linear ordering on $A \times B$.
\end{proof} \end{proof}
\subsection{\verified{Corollary 4P}}% \subsection{\pending{Corollary 4P}}%
\hyperlabel{sub:corollary-4p} \hyperlabel{sub:corollary-4p}
\begin{corollary}[4P] \begin{corollary}[4P]
@ -8320,7 +8321,7 @@ Complete the proof of \nameref{sub:theorem-4k-5}.
\end{proof} \end{proof}
\subsection{\verified{Exercise 4.17}}% \subsection{\pending{Exercise 4.17}}%
\hyperlabel{sub:exercise-4.17} \hyperlabel{sub:exercise-4.17}
Prove that $m^{n+p} = m^n \cdot m^p$. Prove that $m^{n+p} = m^n \cdot m^p$.