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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}
\def\A{\mathop{\cal A}\nolimits}
\def\B{\mathop{\cal B}\nolimits}
\def\N{\mathop{\cal N}\nolimits}
\def\La{\mathop{\cal L}\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\and{\mathop{\wedge}\nolimits}     
\def\implies{\mathop{\to}\nolimits}     
\def\iff{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


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

\begin{enumerate}
\item Given an interpretation $\I = ( \A , h )$ and $\grf,$ $\psi \in
\Form ( \La )$ prove the following.
\begin{description}
\item $\I \models \all x_i \grf$ if and only if for all $a \in A,$
 $\I^{x_i}_a \models \grf.$
\item $\I \models \grf \or \psi$ if and only if $\I \models \grf$ or 
$\I \models \psi.$
\item $\I \models \grf \to \psi$ if and only if whenever $\I \models \grf,$
then $\I \models \psi.$
\item $\I \models \grf \iff \psi$ if and only if ( $\I \models \grf$ iff
$\I \models \psi$).
\end{description}
\item Give an example of formulas $\grf$ and $\psi$ and an $\La$-structure
$M,$ such that $M \models \grf \or \psi,$ but neither $M \models \grf$ nor
$M \models \psi$ holds.
\item Give a formal definition of the set $\var ( \grf )$
of free variables occuring in the formula $\grf \in \Form ( \La ),$
(that is, give a definition based on induction on the complexity of
$\grf$).
\item Let $\grf \in \Form (\La )$ with $\var ( \grf ) = \{ x_0, \ldots x_n \}.$
Show  that for an $\La$-structure $\A,$ $\A \models \grf$ if and only if
$\A \models \all x_0, \ldots x_n\, \grf.$
\item Let $\La = \La_{\PO} = ( \leq )$ be the first order language
for a partial order. Consider the $\La$-structures $\A = ( \N , \leq )$
where $\N$ denotes the natural numbers and $\leq$ the standard patial order
on $\N$ and $\B = ( \N , | )$ where $a|b$ denotes the binary relation
''$a$ divides $b.$'' Let $\grs ( x_0, x_1 )$ be the formula
$$x_0 \leq x_1 \and x_0 \neq x_1 \and \all x_2 (x_0 \leq x_2 \and x_2 \leq x_1
\to x_0 = x_2 \or x_1 = x_2),$$
let $\grf (x_0 )$ be the formula $\exists x_1 \grs ( x_0, x_1)$ and\\
let $\psi (x_0 )$ be the formula $\grf ( x_0 ) \and \all x_1, x_2 
(\grs (x_0, x_1) \and \grs (x_0, x_2) \to x_1 = x_2).$\\
Let $h$ be any instantiation for $\A$ (respectively, $\B$) such that
$h (x_0) \neq 0.$ Evaluate $\grf$ and $\psi$ in $h.$ (Hint: Because $h (x_0)
\neq 0,$ the answer will not depend on $h.$) 
\item Let $\La$ be a first order language with variables
$x_0, x_1, \ldots, x_n, \ldots.$ Prove that every instantiation
$h : \Var ( \La ) \to \A$ extends uniquely to an $\La$-homomorphism
$h'$ from the term algebra of $\La$ with basis $\Var ( \La )$ to the
$\La$-structure $\A.$
\item Let $n$ be a natural number. 
Write a sentence $\grf_n$ in the language of pure equality
such that for every set $M$, $M \models \grf$ if and only 
if $M$ has exactly $n$ elements.
Give an example of a sentence $\grs$ in a first-order language
$\La$ such that whenever $M \models \grs,$ then $M$ is infinite.
\end{enumerate}
\end{document}