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\newtheorem*{solution}{\c C\"oz\"um}

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\begin{document}

\title{Aksiyomatik K\"umeler Kuram\i\ (MAT 340)}
\date{30 May\i s 2013 (f{}inal)}
\author{David Pierce}
\maketitle
\thispagestyle{empty}

\begin{problem}%\mbox{}
\begin{enumerate}[(a)]
\item
\.Iki ordinalin \c carp\i m\i n\i\ tan\i mlay\i n.
\item
Hangi ordinaller, kardinaldir?  (Yani kardinallerin tan\i m\i, nedir?)
\item
\.Iki kardinalin \c carp\i m\i n\i\ tan\i mlay\i n.
\end{enumerate}
\end{problem}

\begin{solution}
\begin{compactenum}[(a)]
\item
$\begin{gathered}[t]
\alpha\cdot0=0,\\
\alpha\cdot\beta'=\alpha\cdot\beta+\alpha,\\
\gamma\text{ limit ise }\alpha\cdot\gamma=\sup_{\xi<\gamma}(\alpha\cdot\xi).
\end{gathered}$
\item
Bir $\kappa$ ordinali bir kardinaldir ancak ve ancak
t\"um $\alpha$ ordinali i\c cin
\begin{equation*}
\kappa\approx\alpha\lto\kappa\leq\alpha.
\end{equation*}
\item
$\kappa\otimes\lambda=\card{\kappa\times\lambda}$.
\end{compactenum}
\end{solution}

\newpage

\begin{problem}
T\"umevar\i m ile t\"um $\alpha$ ordinali i\c cin
\begin{equation*}
1\cdot\alpha=\alpha
\end{equation*}
e\c sitli\u gini kan\i tlay\i n.
\end{problem}

\begin{solution}\mbox{}
\begin{enumerate}
\item
Tan\i mdan $1\cdot0=0$.
\item
$1\cdot\beta=\beta$ ise $1\cdot\beta'=1\cdot\beta+1=\beta+1=\beta'$.
\item
$\gamma$ limit olsun.  E\u ger $\beta<\gamma\lto1\cdot\beta=\beta$ ise
\begin{equation*}
1\cdot\gamma=\sup_{\xi<\gamma}(1\cdot\xi)=\sup_{\xi<\gamma}\xi=\gamma.
\end{equation*}
\end{enumerate}
\end{solution}

\newpage

\begin{problem}
A\c sa\u g\i daki ordinallerin Cantor normal bi\c cimlerini verin.
\begin{enumerate}[(a)]
\item
$\upomega^{\upomega^{\upomega}}+\upomega^{\upomega}+\upomega+1+\upomega^{\upomega}+\upomega+1$
\item
$(\upomega^{\upomega}+\upomega+1)\cdot5$
\item
$(\upomega^{\upomega}+\upomega+1)\cdot\upomega$
\item
$5\cdot(\upomega^{\upomega}+\upomega+1)$
\item
$\upomega\cdot(\upomega^{\upomega}+\upomega+1)$
\item
$(\upomega^{\upomega}+\upomega+1)^{\upomega}$
\end{enumerate}
\end{problem}

\begin{solution}\mbox{}
\begin{enumerate}[(a)]
\item
$\upomega^{\upomega^{\upomega}}+\upomega^{\upomega}\cdot2+\upomega+1$
\item
$\upomega^{\upomega}\cdot5+\upomega+1$
\item
$\upomega^{\upomega+1}$
\item
$\upomega^{\upomega}+\upomega+5$
\item
$\upomega^{\upomega}+\upomega^2+\upomega$
\item
$\upomega^{\upomega^2}$
\end{enumerate}
\end{solution}

\newpage

\begin{problem}
A\c sa\u g\i daki denklemleri sa\u glayan $\xi$ ordinallerini bulun.
\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
$(\aleph_{\alpha})^{\aleph_{\alpha}}=2^{\aleph_{\xi}}$ (kardinal kuvvetler)
\item
$\card{\pow{\aleph_{\upomega+1}}}=2^{\aleph_{\xi}}$ (kardinal kuvvetler)
\item
$\card{\upomega^{\upomega^{\upomega}}+\upomega^{\upomega}+\upomega+75}=\aleph_{\xi}$
\end{enumerate}
\end{problem}

\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}

\end{document}