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\newtheorem{exercise}{Al\i\c st\i rma}
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\begin{document}
\title{Modeller kuram\i\ al\i\c st\i rmalar\i}
\author{David Pierce}
\date{6 Nisan 2017}
\publishers{Matematik B\"ol\"um\"u, MSGS\"U}
\maketitle

\begin{exercise}
  Herhangi form\"ulde sol ayra\c c say\i s\i n\i n,
  sa\u g ayra\c c say\i s\i na e\c sit oldu\u gunu g\"osterin.
\end{exercise}

\begin{exercise}
  $\{<\}$ 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.
$\{<\}$ imzas\i nda,
$T$ teorisine g\"ore niceleyicisiz form\"ule denk \emph{olmayan}
bir $\phi(x,y)$ form\"ul\"un\"u verin.
\end{exercise}

\begin{exercise}
  E\u ger $\str A\preccurlyeq\str C$ ve $\str B\preccurlyeq\str C$ ise
\begin{equation*}
\str A\preccurlyeq\str B
\end{equation*}
elemanter kapsanmas\i n\i\ g\"osterin.
\end{exercise}

\begin{exercise}
  $T_<{}^*$, u\c csuz yo\u gun do\u grusal s\i ralamalar teorisi olsun.
A\c sa\u g\i daki her form\"ul i\c cin
serbest de\u gi\c skenleri ayn\i\ olan ve
$T_<{}^*$ teorisine g\"ore denk olan niceleyicisiz form\"ul\"u bulun.
\begin{enumerate}[a)]
\item 
$\Exists y(x<y)$,
\item 
$\Exists z(x<z\land z<y)$,
\item 
$\Exists z(x<z\land y<z)$,
\item
$\Exists y(x_0=y\land x_1<y\land x_1<x_2)$.
\end{enumerate}
\end{exercise}

\begin{exercise}
  \.Imzas\i\ $\{E\}$ olan $T_{2,\infty}$ teorisinin her $\str A$ modeli i\c cin
$E^{\str A}$, iki s\i n\i fl\i\ denklik ba\u g\i nt\i s\i d\i r,
ve bu ba\u g\i nt\i n\i n her s\i n\i f sonsuzdur.
\begin{enumerate}[a)]
\item 
$T_{2,\infty}$ i\c cin aksiyomlar\i\ yaz\i n.
\item
A\c sa\u g\i daki her form\"ul i\c cin
serbest de\u gi\c skenleri ayn\i\ olan ve
$T_{2,\infty}$ teorisine g\"ore denk olan niceleyicisiz form\"ul\"u bulun.
\begin{enumerate}[i.]
\item 
$\Exists z\lnot(x\mathrel Ez\lor y\mathrel Ez)$,
\item
$\Exists y(x_0\mathrel Ey\land x_1\mathrel Ey\land\lnot\; x_2\mathrel Ey)$. 
\item
$\Exists y(x_0\mathrel Ey\land x_1\mathrel Ey\land\lnot\; x_2\mathrel Ey
\land x_0\neq y)$. 
\end{enumerate}
\end{enumerate}
\end{exercise}

\begin{exercise}
  $\phi$ ve $\psi$ niceleyicisiz ise 
$\Exists x\phi\lto\Forall x\psi$ form\"ul\"un\"u
\"onekli bi\c cimde 
(yani niceleyicilerin \"onde oldu\u gu bi\c cimde) yaz\i n.
\end{exercise}


\begin{exercise}
  Verilen k\"umelerin ve ba\u g\i nt\i lar\i n
  verilen yap\i larda tan\i mlanabildi\u gini g\"osterin.
\begin{enumerate}[a)]
\item
$\{\text{tek say\i lar}\}$, $(\Z,+)$'da
\item
  $\{0\}$, $(\Z,\times)$'da
\item
  $\{0\}$, $(\upomega,<)$'te
\item
  $\{(x,y)\in\upomega^2\colon y-x=2\}$, $(\upomega,<)$'te
\item
$\{\text{bile\c sik say\i lar}\}$, $(\N,\times,1)$'de
\item
  $\{\text{pozitif say\i lar}\}$, $(\Z,\times,1)$'de
\item
  $[3,5]$ aral\i\u g\i, $(\R,+,\times)$'da
\item
  bir grubun, mertebesi $2$ olan elemanlar\i\ k\"umesi, grupta
\item
  bir $G$ grubunun $\cen$ merkezi
  (yani her elemanla de\u gi\c sen elemanlar k\"umesi),
  $G$'de
\item
  $\{(x,y)\in G^2\colon x\cen=y\cen\}$, bir $G$ grubunda
\item
  \"u\c ctane elemana denk olan elemanlar k\"umesi,
  denklik ba\u g\i nt\i s\i ile donat\i lm\i\c s bir k\"umede
\item
  $\{\text{asal say\i lar}\}$, $(\N,{}\mid{})$'de
\item
  $\bigcup_{p\text{ asal}}\{p^k\colon k\in\upomega\}$, $(\N,{}\mid{})$'de
\item
  $\Z$, $(\C,+,\times,\exp)$'te
\end{enumerate}
\end{exercise}

\end{document}
