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\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\I{\mathop{\cal I}\nolimits}
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\def\and{\mathop{\wedge}\nolimits}     
\def\or{\mathop{\vee}\nolimits}     
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\def\ssi{\mathop{\leftrightarrow}\nolimits}     
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\begin{document}
\begin{center} {\bf Problem Set III}\\
Math 609  \end{center}

\begin{enumerate}
\item Give deductions for the axioms for equality.
	\begin{itemize}
   \item $\deduces x \doteq x$
   \item $\deduces x \doteq y \implies y \doteq x$
   \item $\deduces x \doteq y \and  y \doteq z \implies x \doteq z$
 \end{itemize}
\item Give deductions for the following.
 \begin{itemize}
   \item $\forall x \forall y \grf \deduces \forall y \forall x \grf$
   \item $R(x) \deduces \exists y R(y)$
 \end{itemize}
\item (Prenex normal form.) A formula $\grf$ is a {\em prenex formula}
if it has the form $Q_0 x_0\, Q_1 x_1\, \ldots \,Q_n x_n\, \psi$ where $Q_i$ is
$\exists$ or $\forall$ and $\psi$ is quantifier-free. Prove that for any
formula $\grs$ there is a prenex formula $\grf$ such that 
$\deduces \grs \ssi \grf.$
\item Define rigorously, by recursion on the formula $\grf (x),$ when the
term $t$ is substitutable for $x$ in $\grf.$
\item Let $\La$ be a first-order language and $\{ A_i : i \in I \}$ a set
of $\La$-structures. Define the $\La$-structure $\prod_{i \in I} A_i,$ the
Cartesian product of the $A_i.$
\item Classify all the finite structures (up to isomorphism) in the following
cases.
\begin{itemize}
   \item The $\La$-structures where $\La = \La_=$ is the language of pure equality.
   \item The $\La$-structures where $\La = (R)$ and $R$ is a unary relation symbol.
   \item The $\La$-structures that satisfy the formula 
$$E(x,x)\, \and \, (E(x,y) \implies E(y,x)) \, \and \,(E(x,y) \and E(y,z) \implies E(x,z))$$
in the language $\La = (E)$ where $E$ is a binary relation symbol. 
   \item The $\La$-structures that satisfy the formula 
$$(f(x) = f(y) \implies x=y) \and \forall y \exists x \,(f(x) = y)$$
in the language $\La = (f)$ where $f$ is a unary function symbol. 
\end{itemize}
\end{enumerate}

\end{document}