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\cehead{K\"umler kuram\i\ al\i\c st\i rmalar\i}
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 \begin{document}
%\frontmatter
 \title{K\"umeler kuram\i\ al\i\c st\i rmalar\i}
 \author{David Pierce}
\date{\today}
 \publishers{Matematik B\"ol\"um\"u\\
Mimar Sinan G\"uzel Sanatlar \"Universitesi\
\.Istanbul\\
\url{dpierce@msgsu.edu.tr}\\
\url{http://mat.msgsu.edu.tr/~dpierce/}}

\maketitle

\begin{xca}
$\bm A$, $\bm B$, ve $\bm C$ s\i n\i f ise, 
hangi form\"uller a\c sa\u g\i daki s\i n\i flar\i\ tan\i mlar?
\begin{multicols}2
\begin{asparaenum}
\item 
$\bm A\cap\bm B$.
\item 
$\bm A\cup\bm B$.
\item
$(\bm A\cap\bm B)\cup\bm C$.
\item 
$\bm A\setminus\bm B$.
\item 
$\bigcap\bm A$.
\item 
$\bigcup\bm A$.
\item 
$\bigcup\bigcup\bm A$.
\item 
$\pow{\bm A}$.
\end{asparaenum}
  \end{multicols}
\end{xca}

\begin{xca}
A\c sa\u g\i daki k\"umelerin olu\c sturdu\u gu s\i n\i flar\i n\i\ 
tan\i mlayan form\"uller yaz\i n.  
K\i saltmalar kullan\i rsan\i z, onlar\i\ da tan\i mlamal\i s\i n\i z.
\begin{multicols}2
\begin{asparaenum}
\item 
Bo\c s k\"umeler.
\item 
Bo\c s olmayan k\"umeler.
\item
Tek elemanl\i\ k\"umeler.
\item
\.Iki elemanl\i\ k\"umeler.
\item
En \c cok iki elemanl\i\ k\"umeler.
\item
Bo\c s olmayan, en \c cok iki elemanl\i\ k\"umeler.
\item
S\i ral\i\ ikililer.
\item
K\"ume olan iki konumlu ba\u g\i nt\i lar.
\item
K\"ume olan g\"ondermelar.
\item
K\"ume olan ge\c ci\c sli ba\u g\i nt\i lar.
\item
Ge\c ci\c sli k\"umeler.
\item
\.I\c cerilme taraf\i ndan iyis\i ralanm\i\c s k\"umeler.
\item
Ordinaller.
\item
Do\u gal say\i lar.
\item
Kardinaller.
\end{asparaenum}
\end{multicols}
\end{xca}

\begin{xca}
A\c sa\u g\i daki bir ordinaller e\c sitli\u gi her zaman do\u gru ise 
kan\i tlay\i n; de\u gilse bir kar\c s\i t \"ornek verin.
\begin{multicols}2
\begin{asparaenum}
\item
$\alpha+0=\alpha$.
\item
$0+\alpha=\alpha$.
  \item 
$\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma$.
\item
$\alpha+\beta=\beta+\alpha$.
\item
$\alpha\cdot1=\alpha$.
\item
$1\cdot\alpha=\alpha$.
\item
$\alpha\cdot2=\alpha+\alpha$.
\item
$2\cdot\alpha=\alpha+\alpha$.
\item
$\alpha+\beta\cdot\gamma=(\alpha+\beta)\cdot\gamma$.
\item
$\alpha\cdot(\beta\cdot\gamma)=(\alpha\cdot\beta)\cdot\gamma$.
\item
$\alpha\cdot\beta=\beta\cdot\alpha$.
\item
$\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma$.
\item
$(\alpha+\beta)\cdot\gamma=\alpha\cdot\gamma+\beta\cdot\gamma$.
\item
$\alpha^0=0$.
\item
$\alpha^0=1$.
\item
$0^{\alpha}=1$.
\item
$0^{\alpha}=0$.
\item
$\alpha^{\beta}=\beta^{\alpha}$.
\item
$(\alpha+\beta)^2=\alpha^2+\alpha\cdot\beta+\beta\cdot\alpha+\beta^2$.
\item
$\alpha^{\beta+\gamma}=\alpha^{\beta}+\alpha^{\gamma}$.
\item
$(\alpha+\beta)^{\gamma}=\alpha^{\gamma}+\beta^{\gamma}$.
\item
$\alpha^{\beta+\gamma}=\alpha^{\beta}\cdot\alpha^{\gamma}$.
\item
$\alpha^{\beta\cdot\gamma}=(\alpha^{\beta})^{\gamma}$.
\item
$(\alpha\cdot\beta)^{\gamma}=\alpha^{\gamma}\cdot\beta^{\gamma}$.
\item
$\alpha^{(\beta^{\gamma})}=\left(\alpha^{\beta}\right)^{\gamma}$.
  \end{asparaenum}
\end{multicols}
\end{xca}

\begin{xca}
  Cantor normal bi\c cimleri bulun:
  \begin{multicols}2
  \begin{asparaenum}
  \item 
$1+\upomega+\upomega^2+\upomega^3$.
    \item 
$3\cdot(\upomega+4)$.
    \item 
$(\upomega+4)\cdot3$.
\item
$(\upomega^2+\upomega+1)(\upomega^3+\upomega^2+\upomega+1)$.
\item
$(\upomega+5)^2$.
\item
$9^{\upomega+2}$.
\item
$(\upomega+5)^{\upomega+2}$.
\item
$(\upomega^{\upomega})^{\upomega^{\upomega}}$.
\item
$(\upomega^{\upomega^{\upomega}})^{\upomega^{\upomega}}$.
\item
$6^{\upomega^{1330}}$.
\item
$(\upomega^2+\upomega+1)^{\upomega^3+\upomega^2+\upomega+1}$.
  \end{asparaenum}
  \end{multicols}
  \begin{asparaenum}\setcounter{enumi}{11}
  \item
$\upomega^{\upomega^{\upomega}\cdot2+\upomega^{17}}\cdot5
+\upomega^{\upomega^5}\cdot14+\upomega^{\upomega^{\upomega}+\upomega^{17}}\cdot6
+\upomega+317$.
  \end{asparaenum}
\end{xca}

\begin{xca}
Her k\"umenin kardinali, $\aleph_{\alpha}$ veya
$\beth_{\alpha}$ bi\c ciminde yaz\i n.
\begin{multicols}2
\begin{asparaenum}
\item
$\upomega$.
\item
$\upomega^{\upomega}$.
\item
$\sup\left\{\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\right\}$.
\item
Say\i labilir ordinaller k\"umesi.
\item
$\aleph_3\oplus\aleph_5$.
\item
$\aleph_5\oplus\aleph_3$.
\item
$\aleph_3\otimes\aleph_5$.
\item
$\aleph_5\otimes\aleph_3$.
\item
$\aleph_{2\cdot\upomega}\oplus\aleph_{\upomega\cdot2}$.
\item
$(\aleph_2\oplus\aleph_3)\otimes(\aleph_{\upomega}\oplus\aleph_{16})$.
\item
$(\aleph_2\oplus\aleph_3)\otimes(\upomega\oplus\aleph_{16})$.
\item
$(\aleph_2\otimes\aleph_3)\oplus(\aleph_{\upomega}\otimes\aleph_{16})$.
\item
$\aleph_{\upomega^{\upomega}}\oplus\aleph_{\upomega}$.
\item
$\aleph_{\upomega}\oplus\aleph_{\upomega^{\upomega}}$.
\item
$\aleph_{\upomega^{\upomega}}\otimes\aleph_{\upomega}$.
\item
$\R$.
\item
$\pow{\R}$.
\item
$\R$ k\"umesinin sonlu altk\"umeleri k\"umesi.
\item
${}^{\upomega}\R$.
\item
$\R$ k\"umesinin say\i labilir alt\-k\"umeleri k\"umesi.
\item
$\R$ k\"umesinin say\i lamaz alt\-k\"umeleri k\"umesi.
\item
$\left(\aleph_0\right)^{\aleph_0}$.
\item
$\beth_0{}^{\beth_0{}}$.
\item
$\left(\beth_1\right)^{\beth_1}$.
\item
$\left(\aleph_1\right)^{\beth_1}$.
\item
$\sup\{\aleph_0,\aleph_0{}^{\aleph_0},\aleph_0{}^{\aleph_0{}^{\aleph_0}},\dots\}$. 
\item
$\left(\aleph_{\upomega^2\cdot3+\upomega}\right)^{\beth_{\upomega^{\upomega}}}$.
\item
$\beth_{\upomega+1}{}^{\beth_{\upomega}}$.
\item 
$\pow{\beth_{\upomega}}$.
\end{asparaenum}
\end{multicols}
\end{xca}

 \end{document}