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\begin{document}

\title{Modeller Kuram\i\ (MAT 414)\\
Final S\i nav\i\ \emph{\c C\"oz\"umleri}}
\date{27 May\i s 2015}
\author{David Pierce}
%\abstract{}
\maketitle
\thispagestyle{empty}

\begin{problem}
$\{<\}$ imzas\i nda $T_<$,
do\u grusal s\i ralamalar teorisi olsun,
ve $T$, aksiyomlar\i\ $T_<$ teorisinin aksiyomlar\i\ ile
\begin{gather*}
  \Forall x\Exists y\Forall z\bigl(x<y\land(z\leq x\lor y\leq z)\bigr),\\
  \Forall x\Exists y\Forall z\bigl(y<x\land(z\leq y\lor x\leq z)\bigr)
\end{gather*}
olan teori olsun.
\begin{enumerate}[(a)]
\item 
$T$ teorisinin bir modelini verin.
\item
$\{<\}$ imzas\i nda,
$T$ teorisine g\"ore niceleyicisiz form\"ule denk \emph{olmayan}
bir $\phi(x,y)$ form\"ul\"un\"u verin.
\end{enumerate}
\end{problem}

\begin{solution}
  \begin{enumerate}[(a)]
    \item
$(\Z,<)$.
\item
$\Exists z(x<z\land z<y)$.
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
  Bir $\mathscr I$ imsaz\i nda $\str A$ ve $\str B$,
yap\i\ olsun, ve $\str A\included\str B$ 
(yani $\str A$, $\str B$'nin altyap\i s\i)
olsun.
\begin{enumerate}[(a)]
\item 
Tan\i ma g\"ore, ne zaman $\str A\preccurlyeq\str B$ 
(yani $\str A$, $\str B$'nin temel altyap\i s\i)?
\item
$\str C$ de, imzas\i\ $\mathscr I$ olan bir yap\i\ olsun.
E\u ger $\str A\preccurlyeq\str C$ ve $\str B\preccurlyeq\str C$ ise
\begin{equation*}
\str A\preccurlyeq\str B
\end{equation*}
g\"osterin.
\item
$X\included B$ olsun.
$\upomega$'n\i n her $n$ eleman\i\ i\c cin,
imzas\i\ $\mathscr I$ olan her $\phi(x_0,\dots,x_{n-1},y)$ form\"ul\"u i\c cin,
$X^n$ kuvvetinin her $\vec a$ eleman\i\ i\c cin
\begin{equation*}
  \str B\models\Exists y\phi(\vec a,y)
\end{equation*}
durumunda
\begin{align*}
  b_{\phi(\vec a,y)}&\in B,&\str B\models\phi(\vec a,b_{\phi(\vec a,y)})
\end{align*}
olsun.
Bu \c sekilde bir $\bigl(\phi(\vec x,y),\vec a\bigr)\mapsto b_{\phi(\vec a,y)}$ 
g\"ondermesi tan\i mlanm\i\c st\i r.
Bu g\"ondermenin de\u ger k\"umesi $Y$ olsun.  $X\included Y$ g\"osterin.
\end{enumerate}
\end{problem}

\begin{solution}
  \begin{enumerate}[(a)]
    \item
$\upomega$'n\i n her $n$ eleman\i\ i\c cin,
imzas\i\ $\mathscr I$ olan her $\phi(x_0,\dots,x_{n-1})$ form\"ul\"u i\c cin,
$A^n$ kuvvetinin her $\vec a$ eleman\i\ i\c cin
\begin{equation*}
  \str A\models\phi(\vec a)\iff\str B\models\phi(\vec a).
\end{equation*}
\item
Bir $n$ i\c cin $\phi(\vec x)$,
imzas\i\ $\mathscr I$ olan $n$-konumlu form\"ul olsun ve $\vec a\in A^n$ olsun.
O zaman varsay\i mdan
\begin{equation*}
  \str A\models\phi(\vec a)\iff\str C\models\phi(\vec a).
\end{equation*}
Ayr\i ca $\vec a\in B^n$ oldu\u gundan
$\str B\models\phi(\vec a)\iff\str C\models\phi(\vec a)$.
Bu durumda
\begin{equation*}
  \str A\models\phi(\vec a)\iff\str B\models\phi(\vec a).
\end{equation*}
\item
$a\in X$ ise $a$, $b_{a=y}$ olur.
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
Bir $\mathscr I$ imzas\i nda
  $T$,
niceleyicilerin giderilmesine imk\^an veren bir teori olsun.
E\u ger $\str A\models T$, $\str B\models T$, ve $\str A\included\str B$ ise
\begin{equation*}
  \str A\preccurlyeq\str B
\end{equation*}
g\"osterin.
\end{problem}

\begin{solution}
  $\str A\included\str B$ oldu\u gundan her $n$ i\c cin
imzas\i\ $\mathscr I$ olan her niceleyicisiz 
$n$-konumlu $\phi(\vec x)$ form\"ul\"u i\c cin,
$A^n$ kuvvetinin her $\vec a$ eleman\i\ i\c cin
\begin{equation*}
  \str A\models\phi(\vec a)\iff\str B\models\phi(\vec a).
\end{equation*}
$T$, niceleyicilerin giderilmesine imk\^an verdi\u ginden
imzas\i\ $\mathscr I$ olan her $\psi(\vec x)$ form\"ul\"u i\c cin
bir $\phi(\vec x)$ niceleyicisiz form\"ul\"u i\c cin
\begin{equation*}
  T\vdash\Forall{\vec x}(\psi(\vec x)\liff\phi(\vec x)),
\end{equation*}
ve bu durumda, $\str A\models T$ ve $\str B\models T$ oldu\u gundan
\begin{align*}
  \str A\models\psi(\vec a)
&\iff\str A\models\phi(\vec a)\\
&\iff\str B\models\phi(\vec a)\\
&\iff\str B\models\psi(\vec a).
\end{align*}
\end{solution}

\newpage

\begin{problem}
Bu problemde
\begin{equation*}
\Exists{x_0}\dots\;\Exists{x_{n-1}}\bigwedge_{i<j<n}x_i\neq x_j
\end{equation*}
c\"umleleri yararl\i\ olabilir.  
Her $n$ do\u gal say\i s\i\ i\c cin
c\"umlenin k\i saltmas\i\ olarak
$\sigma_n$ kullan\i labilir.
  \begin{enumerate}[(a)]
  \item 
  Her modeli sonsuz olan bir teori var m\i d\i r?
\item
  Her modeli sonlu olan bir teori var m\i d\i r?
\item
  Her modeli sonlu olan 
ama istedi\u gimiz kadar b\"uy\"uk olabilen 
bir teori var m\i d\i r?
  \end{enumerate}
\end{problem}

\begin{solution}
  \begin{enumerate}[(a)]
    \item
Evet, aksiyomlar\i\ 
$\sigma_n$ olan teoridir.
\item
Evet, aksiyomu $\lnot\sigma_2$ olan teoridir.
\item
Hay\i r, \c c\"unk\"u $T$ bu \c sekildeyse $T\cup\{\sigma_n\colon n\in\N\}$ k\"uemesinin her sonlu altk\"umesinin modeli vard\i r,
dolay\i s\i yla T\i k\i zl\i k Teoremine g\"ore
k\"umenin t\"um\"un\"un modeli vard\i r.
  \end{enumerate}
\end{solution}

\end{document}
