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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{25 Nisan 2014}
\author{David Pierce}
\maketitle
\thispagestyle{empty}


\begin{problem}
Cevaplar\i n\i z\i\ k\i saca a\c c\i klay\i n: // 
\emph{Briefly explain your answers:}
  \begin{enumerate}[a)]
  \item 
Her k\"ume bir s\i n\i f m\i d\i r? // \emph{Is every set a class?}
\item
Her s\i n\i f bir k\"ume midir? // \emph{Is every class a set?}
  \end{enumerate}
\end{problem}

\begin{solution}\mbox{}
  \begin{enumerate}[a)]
\item
Evet:  Her $a$ k\"umesi, $\{x\colon x\in a\}$ s\i n\i f\i d\i r.
  \item 
Hay\i r:  Russell Paradoksuna g\"ore $\{x\colon x\notin x\}$ s\i n\i f\i, 
k\"ume de\u gildir.
  \end{enumerate}
\end{solution}

\newpage

\begin{problem}
  \begin{enumerate}[a)]
\item
\textbf{Normal} ordinal i\c slemlerini tan\i mlay\i n. //
\emph{Define the \textbf{normal} ordinal operations.}
  \item 
Ordinal $\xi\mapsto\xi+\xi$ i\c sleminin normal olup olmad\i\u g\i n\i\ 
g\"osterin. // 
\emph{Show whether the ordinal operation $\xi\mapsto\xi+\xi$ is normal.}
  \end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[a)]
\item
E\u ger $\bm F$ kesin artan 
ve her $\alpha$ limiti i\c cin
\begin{equation*}
  \bm F(\alpha)=\sup_{\xi<\alpha}\bm F(\xi)
\end{equation*}
ise, 
$\bm F$ normaldir.
\item    
Normal de\u gildir:
\begin{equation*}
  \sup_{x\in\upomega}(x+x)=\upomega<\upomega+\upomega.
\end{equation*}
  \end{asparaenum}
\end{solution}

\newpage

\begin{problem}
$\left\{\begin{gathered}
\alpha\cdot(\beta+\gamma)=\alpha\cdot\beta+\alpha\cdot\gamma\\
(\alpha+\beta)\cdot\gamma=\alpha\cdot\gamma+\beta\cdot\gamma
  \end{gathered}\right\}$ 
e\c sitliklerinin biri her zaman do\u gru ise kan\i tlay\i n.
// \emph{If one of the equations is always true, prove it.}
\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'.
\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}(\alpha\cdot(\beta+\xi))&&\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}
  \begin{enumerate}[a)]
  \item 
Ge\c ci\c sli k\"umeler s\i n\i f\i n\i\ tan\i mlayan,
serbest de\u gi\c skeni $x$ olan bir form\"ul yaz\i n.
``$\included$'' i\c saretini kullanabilirsiniz. //
\emph{Write down a formula in the free variable $x$ 
defining the class of transitive sets.
You may use the symbol $\included$.}
  \item 
K\"ume olan g\"ondermeler s\i n\i f\i n\i\ tan\i mlayan,
serbest de\u g\-i\c s\-keni $w$ olan bir form\"ul yaz\i n.
``$=$'' i\c saretini ve ``$(x,y)$'' gibi ifadeleri kullanabilirsiniz. //
\emph{Write down a formula in the free variable $z$ 
defining the class of functions that are sets.
You may use the symbol $=$ and expressions like $(x,y)$.}
  \end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[a)]
    \item
$\Forall y(y\in x\lto y\included x)$.
\item
$\begin{aligned}[t]
  &\Forall z\bigl(z\in w\lto\Exists x\Exists y(x,y)=z\bigr)\\
&\land\Forall x\Forall y\Forall z\bigl((x,y)\in w\land(x,z)\in w\lto y=z\bigr).
  \end{aligned}$
  \end{asparaenum}
\end{solution}

\end{document}
