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\begin{document}

\title{\CJK%
{\normalsize 臺北市立大學數學系~105~學年第二學期}\\[4pt]
\textbf{基礎數學}\\
{\large 期末考試題}{\small \S4.5－\S5.3}%
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\begin{Ex}
Let $f:A\to B$ be a function, and $E$ and $F$ be subsets of $B$. Prove that $f^{-1}(E\cap F) = f^{-1}(E) \cap f^{-1}(F)$. \quad (10~points)
\end{Ex}

\begin{Ex}
Prove that the open interval $(0,1)$ is uncountable. \quad (10~points)
\end{Ex}

\begin{Ex}
Prove that the set $\mathbb{N}\times\mathbb{N}$ is denumerable (or countably infinite). \quad (12~points)
\end{Ex}

\begin{Ex}
Let $f:A\to B$ be a one-to-one function, and $C$ and $D$ be subsets of $A$. Prove that $f(C\cap D) = f(C)\cap f(D)$. \quad (14~points)
\end{Ex}

\begin{Ex}
Let $f:A\to B$ be a function, and suppose that $D\subseteq A$ and $E\subseteq B$. Prove that $D = f^{-1}(f(D))$ if and only if $f(A-D) \subseteq B-f(D)$. \quad (14~points)
\end{Ex}

\begin{Ex}
Prove that every subset of countable set is countable. \quad (14~points)
\end{Ex}

\begin{Ex}
Let $\mathcal{F}$ be the set of all binary sequences, i.e., th set of all functions from $\mathbb{N}$ to $\set{0,1}$. Prove that $\mathcal{F}\approx\mathcal{P}(\mathbb{N})$ the power set of $\mathbb{N}$. \quad (14~points)
\end{Ex}

\begin{Ex}
Suppose that $A$ is finite. Prove that $A$ is not equivalent to any of its proper subsets. \quad (16~points)
\end{Ex}
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