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 \begin{document}
 %\frontmatter
 \subtitle{Aksiyomatik K\"umeler Kuram\i}
 \title{Al\i\c st\i rmalar}
 \author{David Pierce\\
 Matematik B\"ol\"um\"u, MSGS\"U}
\date{5 Aral\i k 2019}
 \maketitle\thispagestyle{empty}

 \begin{xca}\label{B}
   Bir $\bm F$ ordinal i\c slemi i\c cin, e\u ger her $\alpha$ i\c cin,
   \begin{equation*}
     \beta<\bm F(\alpha)<\gamma
   \end{equation*}
   ko\c sulunu sa\u glayan
   her $\beta$ ve $\gamma$ i\c cin,
   \begin{equation*}
     \delta<\alpha<\zeta
   \end{equation*}
   ko\c sulunu sa\u glayan
   baz\i\ $\delta$ ve $\zeta$ i\c cin,
   her $\xi$ i\c cin
   \begin{equation*}
     \delta<\xi<\zeta\implies\beta<\bm F(\xi)<\gamma
   \end{equation*}
   ise, o zaman tan\i ma g\"ore $\bm F$ \textbf{s\"ureklidir.}
   $\bm F$ kesin artan olmak \"uzere $\bm F$'nin s\"urekli olmas\i n\i n
   gerek ve yeter bir ko\c sulunun, her $\alpha$ limiti i\c cin
   \begin{equation*}
     \bm F(\alpha)=\sup \bm F[\alpha]
   \end{equation*}
   denkleminin do\u gru olmas\i\ oldu\u gunu g\"osterin.
 \end{xca}

 \begin{xca}
   \"Ornekler varsa, birini verin;
   yoksa olmad\i\u g\i n\i\ kan\i tlay\i n.
   \begin{compactenum}
   \item
     K\"ume olmayan bir s\i n\i f.
   \item
     S\i n\i f olmayan bir k\"ume.
   \item
     Kendisini i\c cermeyen bir k\"ume.
   \item
     Kendisini kapsamayan bir k\"ume.
   \item
     Ordinal olmayan,
     %en az \"u\c c eleman\i\ olan,
     $\in$ taraf\i ndan iyis\i ralanan bir k\"ume.
   \item
     Ordinal olmayan, bo\c s olmayan,
     ge\c ci\c sli bir k\"ume.
   \item
     Elemanlar\i\ ordinal olan, en k\"u\c c\"uk eleman\i\ $1$ olan bir k\"ume.
   \item
     Ordinal olan, en k\"u\c c\"uk eleman\i\ $1$ olan bir k\"ume.
   \item
     Elemanlar\i\ ordinal olan, en k\"u\c c\"uk eleman\i\ olmayan bir k\"ume.
   \item
     Kesin artan, normal olmayan bir ordinaller i\c slemi.
   \item
     S\"urekli olan, kesin artmayan bir ordinaller i\c slemi.
     %(Al\i\c s\-t\i rma \ref{B}'ye bak\i n.)
   \item
     Kesin azalan bir ordinaller i\c slemi.
   \item
     Say\i lamaz bir k\"ume.
   \item
     K\"ume olmayan, say\i labilir bir s\i n\i f.
   \end{compactenum}
 \end{xca}

\begin{xca}
A\c sa\u g\i daki bir ordinaller e\c sitli\u gi her durumda do\u gru ise 
e\c sitli\u gi 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
$2\cdot\alpha=\alpha+\alpha$.
\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
$(\alpha+\beta)\cdot\gamma=\alpha\cdot\gamma+\beta\cdot\gamma$.
\setcounter{mysave}{\value{enumi}}
\end{asparaenum}
\end{multicols}
\begin{asparaenum}
\setcounter{enumi}{\value{mysave}}
\item
$(\alpha+\beta)^2=\alpha^2+2\cdot\alpha\cdot\beta+\beta^2$.
\item
$(\alpha+\beta)^2=\alpha^2+\alpha\cdot\beta+\beta\cdot\alpha+\beta^2$.
%\item$(\alpha+\beta)^3=\alpha^3+\alpha^2\cdot\beta+\alpha\cdot\beta\cdot\alpha+\beta\cdot\alpha^2+\alpha\cdot\beta^2+\beta\cdot\alpha\cdot\beta+\beta^2\cdot\alpha+\beta^3$.
\setcounter{mysave}{\value{enumi}}
\end{asparaenum}
\end{xca}

\begin{xca}
  Cantor normal bi\c cimleri bulun:
\begin{multicols}2
  \begin{asparaenum}
  \item 
$1+\upomega+\upomega^2+\upomega^3$.
  \item 
$1+\upomega^2+\upomega+\upomega^3$.
  \item 
$1+\upomega^3+\upomega+\upomega^2$.
  \item 
$\upomega^3+\upomega+\upomega^2+1$.
    \item 
$3\cdot(\upomega+4)$.
    \item 
$(\upomega+4)\cdot3$.
\item
$(\upomega^2+3)\cdot(\upomega+4)$.
\item
$(\upomega+4)\cdot(\upomega^2+3)$.
\item
$(\upomega^2\cdot5+3)\cdot(\upomega+4)$.
\item
$(\upomega+4)\cdot(\upomega^2\cdot5+3)$.
\setcounter{mysave}{\value{enumi}}
\end{asparaenum}
\end{multicols}
\end{xca}

 \end{document}
