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

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\begin{document}

\title{Aksiyomatik K\"umeler Kuram\i\ (MAT 340)}
\date{11 Nisan 2016}
\author{David Pierce}
\maketitle
\thispagestyle{empty}

\begin{problem}
  Cantor normal bi\c cimlerini bulun.
%(Her \"us te Cantor normal bi\c ciminde olsun.)
  \begin{compactenum}[(a)]
  \item 
$1+\upomega+\upomega^2+\upomega+1$
\vfill
  \item 
$\upomega^2+\upomega+1+\upomega+\upomega^2$
\vfill
\item
$(\upomega+1)\cdot(\upomega+2)$
\vfill
\item
$(\upomega^{\upomega\cdot2}+\upomega^{\upomega+3}\cdot2+\upomega^{16}\cdot3+\upomega+4)\cdot6$
\vfill
\item
$(\upomega^4+\upomega^3\cdot2+\upomega^2\cdot3+\upomega+4)\cdot\upomega^5$
\vfill
\item
$(\upomega^5+28)\cdot\upomega^{\upomega}$
  \end{compactenum}
\end{problem}

\begin{solution}\mbox{}
\begin{compactenum}[(a)]
\item
$\upomega^2+\upomega+1$
\item
$\upomega^2\cdot2$
\item
$\upomega^2+\upomega\cdot2+1$
\item
$\upomega^{\upomega\cdot2}\cdot6+\upomega^{\upomega+3}\cdot2+\upomega^{16}\cdot3+\upomega+4$
\item
$\upomega^9$
\item
$\upomega^{\upomega}$
\end{compactenum}
\end{solution}

\pagebreak

\begin{problem}
A\c sa\u g\i daki ordinal e\c sitli\u gi do\u gru ise kan\i tlay\i n,
de\u gilse bir kar\c s\i t \"ornek verin.
\begin{equation*}
\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma
\end{equation*}
\end{problem}

\begin{solution}
  \begin{asparaenum}[i)]
    \item
$\alpha\cdot(\beta+0)
=\alpha\cdot\beta
=\alpha\cdot\beta+0
=\alpha\cdot\beta+\alpha\cdot0$.
\item
Bir $\gamma$ i\c cin
\begin{equation*}
\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma  
\end{equation*}
olsun.  O zaman
\begin{align*}
 \alpha\cdot(\beta+\gamma')
&=\alpha\cdot(\beta+\gamma)'\\
&=\alpha\cdot(\beta+\gamma)+\alpha\\
&=(\alpha\cdot\beta+\alpha\cdot\gamma)+\alpha\\
&=\alpha\cdot\beta+(\alpha\cdot\gamma+\alpha)\\
&=\alpha\cdot\beta+\alpha\cdot\gamma'.
\end{align*}
\item
$\delta$ limit olsun
ve $\gamma<\delta$ ise 
\begin{equation*}
  \alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma
\end{equation*}
olsun.
$\alpha>0$ varsay\i labilir.
O zaman
\begin{align*}
    \alpha\cdot(\beta+\delta)
&=\alpha\cdot\sup_{\xi<\delta}(\beta+\xi)&&\text{[tan\i m]}\\
&=\sup_{\xi<\delta}\bigl(\alpha\cdot(\beta+\xi)\bigr)&&\text{[$\xi\mapsto\alpha\cdot\xi$ normal]}\\
&=\sup_{\xi<\delta}(\alpha\cdot\beta+\alpha\cdot\xi)&&\text{[hipotez]}\\
&=\alpha\cdot\beta+\sup_{\xi<\delta}(\alpha\cdot\xi)&&\text{[$\xi\mapsto\alpha\cdot\beta+\xi$ normal]}\\
&=\alpha\cdot\beta+\alpha\cdot\delta.&&\text{[tan\i m]}
\end{align*}
  \end{asparaenum}
\end{solution}

\newpage


\begin{problem}
A\c sa\u g\i daki ordinal e\c sitli\u gi do\u gru ise kan\i tlay\i n,
de\u gilse bir kar\c s\i t \"ornek verin.
\begin{equation*}
(\alpha+\beta)\cdot\gamma=\alpha\cdot\gamma+\beta\cdot\gamma
\end{equation*}
\end{problem}

\begin{solution}
$\begin{aligned}[t]
(1+1)\cdot\upomega&=2\cdot\upomega=\upomega\\
1\cdot\upomega+1\cdot\upomega&=\upomega+\upomega=\upomega\cdot2>\upomega	
\end{aligned}$
\end{solution}

\pagebreak


\newpage
\begin{problem}
A\c sa\u g\i daki kan\i t nerede yanl\i\c st\i r?
Anlat\i n.
\begin{compactenum}[I.]
\item
\.Ilk olarak
\begin{align}
0'
&=1\\
&=1+0.
\end{align}
\item
\c Simdi
\begin{equation}
\beta'=1+\beta
\end{equation}
olsun.
O zaman
\begin{align}
	\beta'+1
	&=(1+\beta)+1\\
	&=1+(\beta+1)\\
	&=1+\beta'.
\end{align}
\item
Son olarak $\gamma$ limit olsun, 
ve $\alpha<\gamma$ durumunda
\begin{equation}
\alpha'=1+\alpha
\end{equation} 
olsun.
O zaman
\begin{align}
	\gamma'
&=\sup_{\xi<\gamma}(\xi')\\
&=\sup_{\xi<\gamma}(1+\xi)\\
&=1+\gamma.
\end{align}
\end{compactenum}
Sonu\c c olarak her $\alpha$ i\c cin
\begin{equation}
\alpha'=1+\alpha.
\end{equation}
\end{problem}

\begin{solution}
Sat\i r (8) yanl\i\c st\i r,
\c c\"unk\"u 
$\xi\mapsto\xi'$ i\c slemi normal de\u gildir,
ama
$\sup_{\xi<\gamma}(\xi')=\gamma$.
\end{solution}

\end{document}
