Enderton (logic). Add exercise 1.3 theorems and exercise prompts.

finite-set-exercises
Joshua Potter 2023-08-23 14:47:51 -06:00
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TODO TODO
\end{proof} \end{proof}
\section{A Parsing Algorithm}%
\hyperlabel{sec:parsing-algorithmm}
\subsection{\unverified{Lemma 13A}}%
\hyperlabel{sub:lemma-13a}
\begin{lemma}[13A]
Every wff has the same number of left as right parentheses.
\end{lemma}
\begin{proof}
Refer to \nameref{sub:balanced-parentheses}.
\end{proof}
\subsection{\sorry{Lemma 13B}}%
\hyperlabel{sub:lemma-13b}
\begin{lemma}[13B]
Any proper initial segment of a wff contains an excess of left parentheses.
Thus no proper initial segment of a wff can itself be a wff.
\end{lemma}
\begin{proof}
TODO
\end{proof}
\section{Exercises 1}% \section{Exercises 1}%
\hyperlabel{sec:exercises-1} \hyperlabel{sec:exercises-1}
@ -1930,4 +1956,91 @@
\end{proof} \end{proof}
\subsection{\sorry{Exercise 1.3.1}}%
\hyperlabel{sub:exercise-1.3.1}
Rewrite the tautologies in the "selected list" at the end of Section 1.2, but
using the conventions of the present section to minimize the number of
parentheses.
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.2}}%
\hyperlabel{sub:exercise-1.3.2}
Give an example of wffs $\alpha$ and $\beta$ and expressions $\gamma$ and
$\delta$ such that $(\alpha \land \beta) = (\gamma \land \delta)$ but
$\alpha \neq \gamma$.
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.3}}%
\hyperlabel{sub:exercise-1.3.3}
Carry out the argument for \nameref{sub:lemma-13b} for the case of the
operation $\mathcal{E}_{\neg}$.
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.4}}%
\hyperlabel{sub:exercise-1.3.4}
Suppose that we modify our definition of wff by omitting all \textit{right}
parentheses.
Thus instead of
$$((A \land (\neg B)) \Rightarrow (C \lor D))$$
we use $$((A \land (\neg B \Rightarrow (C \land D.$$
Show that we still have unique readability (i.e., each wff still has only one
possible decomposition).
\textit{Suggestion}: These expressions have the same number of parentheses as
connective symbols.
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.5}}%
\hyperlabel{sub:exercise-1.3.5}
The English language has a tendency to use two-part connectives: "both
$\ldots$ and $\ldots$" "either $\ldots$ or $\ldots$" "if $\ldots$, then
$\ldots$."
How does this affect unique readability in English?
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.6}}%
\hyperlabel{sub:exercise-1.3.6}
We have given an algorithm for analyzing a wff by constructing its tree from
the top down.
There are also ways of constructing the tree from the bottom up.
This can be done by looking through the formula for innermost pairs of
parentheses.
Give a complete description of an algorithm of this sort.
\begin{answer}
TODO
\end{answer}
\subsection{\sorry{Exercise 1.3.7}}%
\hyperlabel{sub:exercise-1.3.7}
Suppose that left and right parentheses are indistinguishable.
Thus, instead of $\alpha \lor (\beta \land \gamma))$ we have
$|\alpha \lor |\beta \land \gamma||$.
Do formulas still have unique decomposition?
\begin{answer}
TODO
\end{answer}
\end{document} \end{document}