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 \begin{document}
 \subtitle{Aksiyomatik K\"umeler Kuram\i}
 \title{Al\i\c st\i rmalar}
 \author{David Pierce\\
 Matematik B\"ol\"um\"u, MSGS\"U}
\date{2 Ocak 2020}
 \maketitle\thispagestyle{empty}



\begin{xca}
  Cantor normal bi\c cimleri bulun:
\begin{enumerate}[a)]
%\item
%$\upomega^{\upomega}\cdot2+\upomega^{\upomega+1}+\upomega^5\cdot8+\upomega^{\upomega}+\upomega^5+\upomega\cdot2$
\item
$\upomega^{\upomega^{\upomega}\cdot2+\upomega^{17}}\cdot5
+\upomega^{\upomega^5}\cdot14+\upomega^{\upomega^{\upomega}+\upomega^{17}}\cdot6
+\upomega+317$
%\item
%$(\upomega^2+\upomega+1)\cdot(\upomega^3+\upomega^2+\upomega+1)$
%\item
%$(\upomega^2\cdot4+\upomega\cdot2+5)\cdot(\upomega^3\cdot16+\upomega^2\cdot7+\upomega\cdot8+87)$
\item
$(\upomega^2\cdot4+\upomega\cdot2+5)\cdot(\upomega^{\upomega\cdot3}\cdot16+\upomega^2\cdot7+\upomega\cdot8+87)$
\item
$(\upomega^{\upomega\cdot2}\cdot4+\upomega\cdot2+5)\cdot(\upomega^{\upomega\cdot3}\cdot16+\upomega^2\cdot7+\upomega\cdot8+87)$
\item
$(\upomega^{\upomega\cdot2}\cdot4+\upomega\cdot2+5)\cdot(\upomega^{\upomega^3}\cdot16+\upomega^2\cdot7+\upomega\cdot8+87)$
\setcounter{mysave}{\value{enumi}}
\end{enumerate}
\vfill
\begin{multicols}2
\begin{enumerate}[a)]
\setcounter{enumi}{\value{mysave}}
\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{enumerate}
  \end{multicols}
\end{xca}
\vfill

\begin{xca}
  \c C\"oz\"un:
  \begin{enumerate}[a)]
  \item 
$\xi+\upomega^2+\eta=15+\upomega^2+16$
  \item 
    $\xi\cdot\upomega+\eta\cdot\upomega=(\xi+\eta)\cdot\upomega$ 
  \end{enumerate}
\end{xca}

\begin{solution}
\mbox{}
  \begin{enumerate}[a)]
  \item
    $15+\upomega^2+16=\upomega^2+16$, dolay\i s\i yla
    \begin{equation*}
      \xi<\upomega^2\land\eta=16.
    \end{equation*}
  \item
    E\u ger $\alpha>0$ ise,
    o zaman baz\i\ $\beta$, $n$, ve $\gamma$ i\c cin
    \begin{align*}
      0&<n<\upomega,&
      \gamma&<\upomega^{\beta},&
      \alpha&=\upomega^{\beta}\cdot n+\gamma.
    \end{align*}
    Bu durumda
    \begin{equation*}
      \deg(\alpha)=\beta
    \end{equation*}
    olsun.
    O zaman
    \begin{multline*}
      \xi\cdot\upomega+\eta\cdot\upomega
      =\\
      \begin{cases}
        \upomega^{\deg(\xi)+1}+\upomega^{\deg(\eta)+1},&\deg(\xi)\geq\deg(\eta)\text{ ise},\\
        \upomega^{\deg(\eta)+1},&\deg(\xi)<\deg(\eta)\text{ ise},
      \end{cases}
    \end{multline*}
    ve ayr\i ca
    \begin{equation*}
      (\xi+\eta)\cdot\upomega
      =
      \begin{cases}
        \upomega^{\deg(\xi)+1},&\deg(\xi)\geq\deg(\eta)\text{ ise},\\
        \upomega^{\deg(\eta)+1},&\deg(\xi)<\deg(\eta)\text{ ise}.
      \end{cases}
    \end{equation*}
    B\"oylece denklemin tam \c c\"oz\"um\"u,
    \begin{equation*}
      \xi=0\lor\eta=0\lor\deg(\xi)<\deg(\eta).
    \end{equation*}
  \end{enumerate}
\end{solution}

\newpage

\begin{xca}
  \c C\"oz\"un.
  \begin{enumerate}[a)]
  \item 
$\aleph_1\oplus\aleph_{\xi}=\aleph_3$
\item
$\aleph_{\xi}\otimes\aleph_{\upomega}=\aleph_{\upomega}$
\item
$(\aleph_{\upomega}\oplus\aleph_{\upomega^2})\otimes\aleph_{\upomega\cdot3}=\aleph_{\xi}$
\item
$(\aleph_{\alpha})^{\aleph_{\alpha}}=2^{\aleph_{\xi}}$
\item
$\card(\pow{\aleph_{\xi}})=2^{\aleph_{\upomega+1}}$
\item
$\card(\upomega^{\upomega^{\upomega}}+\upomega^{\upomega}+\upomega+75)=\aleph_{\xi}$
  \end{enumerate}
\end{xca}

\begin{solution}\mbox{}
\begin{enumerate}[(a)]
\item
$\xi=3$
\item
$\xi\leq\upomega$
\item
$\xi=\upomega^2$
\item
$\xi=\alpha$
\item
$\xi=\upomega+1$
\item
$\xi=0$
\end{enumerate}
\end{solution}

\vfill
\begin{xca}
Her k\"umenin kardinali, $\aleph_{\alpha}$ veya
$\beth_{\alpha}$ bi\c ciminde yaz\i n.
\begin{compactenum}[a)]
\item
Say\i labilir ordinallerin olu\c sturdu\u gu k\"ume
\item
$\R$'nin sonlu altk\"umelerinin olu\c sturdu\u gu k\"ume
\item
$\R$'nin say\i labilir alt\-k\"umelerinin olu\c sturdu\u gu k\"ume
\item
$\R$'nin say\i lamaz alt\-k\"umelerinin olu\c sturdu\u gu k\"ume
\item
  $\sup\{\aleph_0,\aleph_0{}^{\aleph_0},\aleph_0{}^{\aleph_0{}^{\aleph_0}},
  \aleph_0{}^{\aleph_0{}^{\aleph_0{}^{\aleph_0}}},\dots\}$ 
\setcounter{mysave}{\value{enumi}}
\end{compactenum}
\vfill
\begin{multicols}2
\begin{enumerate}[a)]
\setcounter{enumi}{\value{mysave}}
\item
$\sup\left\{\upomega,\upomega^{\upomega},\upomega^{\upomega^{\upomega}},\dots\right\}$
\item
$\aleph_3\oplus\aleph_5$
\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
$\aleph_{\upomega}\oplus\aleph_{\upomega^{\upomega}}$
\item
$\aleph_{\upomega^{\upomega}}\otimes\aleph_{\upomega}$
\item
$\pow{\R}$
\item
${}^{\upomega}\R$
\item
$\left(\aleph_0\right)^{\aleph_0}$
\item
$\left(\beth_0\right)^{\beth_0}$
\item
$\left(\beth_1\right)^{\beth_1}$
\item
$\left(\aleph_1\right)^{\beth_1}$
\item
$\left(\aleph_{\upomega^2\cdot3+\upomega}\right)^{\beth_{\upomega^{\upomega}}}$
\item
$\left(\beth_{\upomega+1}\right)^{\beth_{\upomega}}$
\item 
$\pow{\beth_{\upomega}}$
\end{enumerate}
\end{multicols}
\end{xca}



 \end{document}
