\documentstyle{article}

\textwidth=6.25in
\textheight=8.5in
\oddsidemargin=0.25in
\evensidemargin=0.25in
\topmargin=0in
\renewcommand{\baselinestretch}{1.025}

\def\Sent{\mathop{\rm Sent}\nolimits}
\def\Form{\mathop{\rm Form}\nolimits}
\def\Var{\mathop{\rm Var}\nolimits}
\def\VS{\mathop{\rm VS}\nolimits}
\def\var{\mathop{\rm var}\nolimits}
\def\tp{\mathop{\rm tp}\nolimits}
\def\eq{\mathop{\rm eq}\nolimits}
\def\PO{\mathop{\rm PO}\nolimits}
\def\Th{\mathop{\rm Th}\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\F{\mathop{\cal F}\nolimits}
\def\Po{\mathop{\cal P}\nolimits}
\def\G{\mathop{\cal G}\nolimits}
\def\La{\mathop{\cal L}\nolimits}
\def\x{\mathop{\bf x}\nolimits}
\def\y{\mathop{\bf y}\nolimits}
\def\grf{\mathop{\varphi}\nolimits}     
\def\grs{\mathop{\sigma}\nolimits}     
\def\gro{\mathop{\omega}\nolimits}     
\def\gre{\mathop{\eta}\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\deduces{\mathop{\vdash}\nolimits}     
\def\ssi{\mathop{\leftrightarrow}\nolimits}     
\def\all{\mathop{\forall}\nolimits}     
\def\proves{\mathop{\vdash}\nolimits}  
   


\begin{document}
\begin{center} {\bf Final Exam}\\
Math 609  \end{center}


Answer the following questions as clearly and completely as humanly possible. 

\begin{enumerate}
\item State and prove the Soundness Theorem of propositional logic.
\item Let $\La$ be a first-order language. Give the definition of a term
of $\La.$ Give the definition of an atomic formula of $\La.$ Give an
example of a language $\La$ and an atomic formula in $\La$ which is a
sentence. 
\item State the \L o\'s-Vaught Test.
\item Let $F$ be an infinite field and denote by VS($F$) the theory of
vector spaces over $F$ in the language of vector spaces over $F.$ Prove
that this theory is complete.
\item Let $\La = (f)$ be the first-order language whose signature consists
of just one unary function symbol $f.$ Write a sentence $\grf$ (respectively, $\psi$)
that expresses that $f$ is injective (respectively, surjective).
\item Let $\La$ be as in the previous exercise and consider the theory 
$$\{ \grf \wedge \psi,\, \forall x\, (f (x) \neq x),\, \forall x\, (f^2 (x) \neq x), 
\ldots, \forall x\, (f^n (x) \neq x) ,\, \ldots \}.$$
Prove that this theory is consistent, complete and not finitely axiomatizable.
\item Express in the language $\La = (<)$ the axioms for a dense linear order
without endpoints. Prove that this theory is $\aleph_0$-categorical.
\item Let $f : N \to N$ be a total function on the natural numbers whose
graph is definable in the language of arithmetic by a $\,\Sigma_1$ formula.
Prove that the graph of $f$ is then definable by a $\,\Delta_1$ formula.
\item Give a definition by primitive recursion of the function
$$<x,y> = \frac{1}{2} (x+y)(x+y+1) + x.$$
\end{enumerate}


\end{document}