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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Foundations \hspace{.7in} Due 2/13/17} \end{center}

\noindent Four of these problems will be graded (our choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}

\item Prove that if $A$ has $n$ elements, then $\mathcal{P}(A)$ has {$2^n$} elements. [Hint: Induction!]
\item For any sets $A$ and $B$, $(A \bigcup B)' = A' \bigcap B'$.
\item For any sets $A, B$, and $C$, $A \bigcap (B \bigcup C) = (A \bigcap B) \bigcup (A \bigcap C)$.
\item Score at least 7 out of 8 on the Set Theory Gateway on WeBWorK.
\item Suppose $A \subseteq B$.  Show that $A \cup C \subseteq B \cup C$.
\item Show that $A - (B \cap A) = (A - B) \cap A$.
\item Show that \[\left(  \bigcup_{i \in I} A_i \right)^\prime = \bigcap_{i \in I} A_i^\prime \]
\item Show that \[A \cap \bigcup_{i \in I} B_i = \bigcup_{i \in I} \left( A \cap B_i \right)\]

\end{enumerate}

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