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\begin{document}

\begin{center}
\begin{Large}
{\bf Math 2001 - Assignment 7}
\end{Large}

\bigskip

Due March 4, 2026
\end{center}

\thispagestyle{empty}

\bigskip

\begin{enumerate}

%\item\cite[Exercise 4.5.2]{Ma:MT}
% Let $T$ be the theory of $(\Z,s)$ where $s(x) = x+1$ for $x\in\Z$.

\item
  Which of the following are types of the respective structures?
  For those which are, determine the elements in that structure that realize them. 
\begin{enumerate}
\item $\{ \exists y\ xy = 1 \}$ for $(\N,\cdot,1)$, 
\item $\{x>0, x> 1, x>2,\dots\}$ for $(\Z,<)_\N$,
\item $\{x>0, x< 1 \}$ for $(\Z,<)_\N$,
\item $\{x>0, x< 1, x < \frac{1}{2}, x < \frac{1}{3}, \dots \}$ for $(\Q,<)_\Q$.
\end{enumerate}
\end{enumerate}

\bigskip
  
 For the following fix an $\LL$-theory $T$.
\begin{itemize}
\item Let $S_n(T)$ be the set of all complete $n$-types of $T$.
\item
 For an $\LL$-formula $\phi(\bar{x})$, let $[\phi]$ denote the set of complete $n$-types of $T$ that contain $\phi$.
 Recall that these are the basic open sets of the Stone topology on $S_n(T)$.
\end{itemize}


\bigskip

\begin{enumerate}
% TZ, Lemma 4.2.1 
\item[(2)]
 Show that $[\phi] = [\psi]$ iff $\phi$ and $\psi$ are \emph{equivalent modulo} $T$, i.e.,
\[ T\models \forall \bar{x}\ (\phi(\bar{x}) \leftrightarrow \psi(\bar{x})). \]
   
\item[(3)]
% TZ, Lemma 4.2.1 
 Complete the proof from class that the sets $[\phi]$ are closed under the Boolean operations:
\begin{enumerate}
\item $[\phi] \cup [\psi] = [\phi\vee\psi]$
\item $[\phi] \cap [\psi] = [\phi\wedge\psi]$
\item $S_n(T) \setminus [\phi]  = [\neg\phi]$
\item $S_n(T) = [\top]$
\item $\emptyset = [\bot]$
\end{enumerate}  

\item[(4)]
% TZ, Lemma 4.2.3
  (For students who have taken some basic topology)
  
 Show that every clopen subset of $S_n(T)$ is of the form $[\phi]$ for some formula $\phi$.

 Hint: For a clopen $c$, both $c$ and its complement $S_n(T)\setminus c$ are open, hence unions of basic opens sets.
 Use this to get an open cover of $S_n(T)$, apply compactness and (3).  
\end{enumerate}


%\begin{thebibliography}{1}
%\bibitem{En:MIL}
% Enderton. A Mathematical Introduction to Logic. Harcourt/Academic
%Press, 2nd edition, 2001. 

%\bibitem{Ma:MT}
%  Marker.
%\newblock Model Theory: An Introduction.
%\newblock Springer, 2002.

%\bibitem{TZ:CMT}
%  Tent, Ziegler. A Course in Model Theory. Lecture Notes in Logic, Cambridge, 2012. 
%\end{thebibliography}

\end{document}





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