\documentstyle{article}

\textwidth=6.25in
\textheight=8.0in
\oddsidemargin=0.25in
\evensidemargin=0.25in
\topmargin=.5in
\renewcommand{\baselinestretch}{1.025}

\def\Sent{\mathop{\rm Sent}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\grS{\mathop{\Sigma}\nolimits}     
\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}     


\begin{document}
\begin{center} {\bf Problem Set I}\\
Math 609  \end{center}

\begin{enumerate}
\item Let $\grf$ be a sentence of the propositional language ${\cal S}$
and denote by $\ell (\grf)$ (respectively, $r (\grf)$) the number of left
(respectively, right) parentheses in $\grf.$ Prove that for all sentences
$\grf,$ $\ell (\grf ) = r (\grf ).$

\item  Let $\grf$ be a sentence of length $n.$ For each $k$ such that
$1 \leq k \leq n,$ let $\ell (\grf , k)$ (respectively, $r (\grf, k)$)
denote the number of left (respectively, right) parentheses among the first
$k$ symbols of $\grf.$ Prove that for all $k$ such that $1 \leq k < n,$
$\ell (\grf , k)  >  r (\grf, k).$

\item Let ${\cal S}$ be a propositional language. 
Define $S_k$ for every natural number $k$ by recursion as follows:\\
i. $S_0 = {\cal S},$\\
ii. $S_{n+1} = S_n \cup \{ ( \neg \grf ) : \grf \in S_n \} \cup  \{ (\grf \and \psi) : \grf, \psi \in S_n  \}.$\\
Prove that $\Sent ({\cal S}) = \bigcup_k S_k.$

\item A sentence $\grf$ of ${\cal S}$ is called a {\em literal} if it is either
a sentence symbol $p$ or the negation $(\neg p)$ of a sentence symbol. Show that
every sentence $\grf$ of ${\cal S}$ is equivalent (semantically) to a finite
disjunction of sentences each of which is a finite conjuction of literals. This is
called the {\em disjunctive normal form} of $\grf.$ Prove also that $\grf$ is
equivalent to a finite conjunction of sentences each of which is a finite disjunction
of literals. 

\item A connective $c$ is a function $c : \{ t, f \}^n \to \{ t, f\}.$
A set of connectives $C$ is said to be {\em adequate} for propositional logic if every
connective may be represented in the propositional whose connectives are given by $C.$
Show that $\{ \and, \neg \}$ are adequate for propositional logic. 
Give an example of a binary connective $c$ such that the singleton set $\{ c \}$ is
adequate for propositional logic.

\item Show, by giving a deduction, that $\{ \neg p, p \or q \} \proves q.$

\item A subset $\grS \subseteq \Sent ({\cal S})$ is {\em complete} if for every
sentence $\grs$ of ${\cal S},$ exactly one of $\grS \proves \grs$ and 
$\grS \proves \neg \grs$ holds. Show that for any set $\grS$ of sentences the
following are equivalent: \begin{itemize}
    \item The set of consequences of $\grS$ is maximal consistent.
    \item The theory $\grS$ is complete.
    \item The theory $\grS$ has exactly one model.
\end{itemize}

\item {\bf Interpolation theorem.} Assume that $\grf \models \psi.$ Show that either
(i) $\grf$ is refutable, (ii) $\psi$ is valid or (iii) there exists a sentence
$\theta$ such that $\grf \models \theta$ and $\theta \models \psi$ and every 
sentence symbol that occurs in $\theta$ occurs in both $\grf$ and $\psi.$

\item Prove the following. (In the latter two, assume that ${\cal S}$ is countable)
\begin{itemize}
\item For every finite set $K$ of models, there is a set $\grS$ of sentences such
that $K$ is the set of all models of $\grS.$
\item Give an example of a set $\grS$ of sentences such that the set of models
of $\grS$ is countably infinite.
\item Give an example of a countable set of models which cannot be represented as 
the set of models of some set $\grS$ of sentences.
\end{itemize}

\end{enumerate}

\end{document}