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\begin{document}

\title{Aksiyomatik K\"umeler Kuram\i\ (MAT 340)}
\subtitle{\c C\"oz\"umler}
\date{19 Aral\i k 2019}
\author{David Pierce}
\maketitle
\thispagestyle{empty}

\begin{problem}
  Verilen ordinal i\c slemler s\"urekli midir?  K\i saca a\c c\i klay\i n.
  \begin{compactenum}[(a)]
  \item
    $\xi\mapsto\upomega^2+\xi$
    \vfill
  \item
    $\xi\mapsto\upomega\cdot\xi+1$
    \vfill
  \end{compactenum}
\end{problem}

\begin{solution}\mbox{}
  \begin{enumerate}[(a)]
  \item
    S\"ureklidir \c c\"unk\"u normaldir.
    (Her $\alpha$ i\c cin $\xi\mapsto\alpha+\xi$ normaldir.)
  \item
    S\"urekli de\u gildir \c c\"unk\"u
    \begin{equation*}
      \sup_{x<\upomega}(\upomega\cdot x+1)=\upomega\cdot\upomega
      <\upomega\cdot\upomega+1.
    \end{equation*}
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
  $\upomega$'da her $n$ ve her pozitif $k$ i\c cin
  \begin{equation*}
    (\upomega\cdot k+n)^2=\upomega^2\cdot k+\upomega\cdot k\cdot n+n.
  \end{equation*}
  \begin{enumerate}[(a)]
  \item
    A\c sa\u g\i daki kan\i t nerede yanl\i\c st\i r?
  \begin{quote}
    E\u ger $k=1$ ise, o zaman iddia do\u grudur \c c\"unk\"u
    \begin{align}
      (\upomega\cdot 1+n)^2
      &=(\upomega+n)^2\\
      &=(\upomega+n)\cdot(\upomega+n)\\
      &=(\upomega+n)\cdot\upomega+(\upomega+n)\cdot n\\
      &=\upomega^2+\upomega\cdot n+n\\
      &=\upomega^2\cdot1+\upomega\cdot1\cdot n+n.
    \end{align}
    \c Simdi $k=m$ olmak \"uzere iddia do\u gru olsun.  O zaman
    \begin{align}\notag
      &\phantom{{}={}}(\upomega\cdot m'+n)^2\\
      &=((\upomega\cdot m+\upomega)+n)^2\\
      &=(\upomega\cdot m+(\upomega+n))^2\\
      &=\upomega^2\cdot m+\upomega\cdot m\cdot(\upomega+n)+\upomega+n\\
      &=\upomega^2\cdot m+\upomega^2+\upomega\cdot m\cdot n+\upomega+n\\
      &=\upomega^2\cdot m'+\upomega\cdot m'\cdot n+n.
    \end{align}
  \end{quote}
\item
  Do\u gru bir kan\i t verin.
  (T\"umevar\i m kullan\i lmayabilir.)
  \end{enumerate}
\end{problem}

\newpage

\begin{solution}\mbox{}
  \begin{enumerate}[(a)]
  \item
    (8) ve (10) yanl\i\c st\i r.
  \item
    $\begin{aligned}[t]
      (\upomega\cdot k+n)^2
      &=(\upomega\cdot k+n)\cdot(\upomega\cdot k+n)\\
      &=(\upomega\cdot k+n)\cdot\upomega\cdot k+(\upomega\cdot k+n)\cdot n\\
      &=\upomega^2\cdot k+\upomega\cdot k\cdot n+n.
    \end{aligned}$
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
  Cantor normal bi\c cimlerini bulun.
%(Her \"us te Cantor normal bi\c ciminde olsun.)
  \begin{compactenum}[(a)]
  \item 
    $3+\upomega\cdot2+\upomega^2+\upomega\cdot2+3$
    \vfill
  \item 
$\upomega^2+\upomega+\upomega^2+\upomega+\upomega^2$
  \end{compactenum}
\end{problem}
\vfill
\begin{solution}\mbox{}
\begin{enumerate}[(a)]
\item
$\upomega^2+\upomega\cdot2+3$
\item
$\upomega^2\cdot3$
\end{enumerate}
\end{solution}
\vfill
\begin{problem}
  $\in$ taraf\i ndan iyis\i ralanm\i\c s olan,
  ge\c ci\c sli olmayan,
    \"u\c c elemanl\i\ bir k\"ume yaz\i n.
\end{problem}
\vfill
\begin{solution}
  En basit \"ornek
  $\{1,2,3\}$.
\end{solution}
\vfill

\end{document}