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\begin{document}
\noindent

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\noindent\framebox[\linewidth]{\shortstack[c]{
\Large{\textbf{Lab05-Cardinality}}\vspace{1mm}\\
�������ѧ���ۿκ�ϰ�⣬��ʦ�������h��2016 �＾ѧ��}}
\begin{center}
\footnotesize{\color{red}��ע����\quad ������\_\_\_\_\_\_\_\_\_\_\_\_ \quad ѧ�ţ�\_\_\_\_\_\_\_\_\_\_\_ \quad �༶��\_\_\_\_\_\_\_\_\_\_\_}

\footnotesize{\color{blue}$*$ ���Ӱ���ҵ��ֱ���ϴ����γ���վ}
\end{center}

\begin{enumerate}
\item ��$\omega $��ʾ��Ȼ������$\{0,1,2,3,\cdots \}$,֤����ʽ$f(m,n)=2^m(2n+1)-1$�����˴�$\omega \times \omega$��$\omega$��һ��˫���ϵ��{\color{blue}(���������Ϊ\emph{coding} $\pi$����һ�ֹ㷺ʹ�õĽ�$\mathbb{N}\times\mathbb{N}$�е�Ԫ��$(m,n)$ӳ�䵽$\mathbb{N}$�ϵ�ӳ�䷽ʽ������ G\"{o}del Coding.)}
    
\item ��$m < n \in \mathbb{N}$��֤��ʵ������$[0,1]$��$(m,n)$���ơ�

\item ������������$A$��$B$����$\text{card}~A=a$��$\text{card}~B=b$��
\begin{enumerate}    
\item $\text{card}~{}^{A}B=?$
\item ֤����$b \ge 2$ʱ��������˫��$f: A \rightarrow {}^{A}B$��
\item ��$A=\omega$��$B=\{0,1\}$����֪$\text{card}~\omega=\aleph_{0}$����$\text{card}~{}^{\omega}\{0,1\}=?$
\end{enumerate}

\end{enumerate}
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\end{document}
