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\begin{center}\textbf{Problem Set 1 \hspace{.7in} Set Theory \& Topology \hspace{.7in} Due 1/22/16} \end{center}

\noindent You are expected to do the following problems to a high standard (i.e., at least well enough to be published in a textbook) for full credit.

\begin{enumerate}
\item Let $\{A_\alpha : \alpha \in \Lambda \}$ be an indexed collection of sets and let $\Delta$ be a nonempty subset of $\Lambda$.  Prove the following statements:

\begin{enumerate}
\item $\bigcup \{A_\alpha : \alpha \in \Delta \} \subseteq \bigcup \{A_\alpha : \alpha \in \Lambda \}$
\item $\bigcap \{A_\alpha : \alpha \in \Lambda \} \subseteq \bigcap \{A_\alpha : \alpha \in \Delta \}$
\end{enumerate}

\item Let $f:X \to Y$ and $g: Y \to X$ be one-to-one and onto functions.  Then $g = f^{-1}$ iff $f \circ g = \mathrm{id}_Y$ and $g \circ f = \mathrm{id}_X$.

\end{enumerate}

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