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\newtheorem{thm}{Theorem}
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\lhead{MATH 311-01}
\chead{Problem of the Day}
\rhead{YOUR NAME HERE}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{\textbf{due November 25, 2013}}
\begin{document}
\begin{enumerate}
\item \emph{For sets $A$ and $B$, prove that if $A$ is uncountable
    (i.e. $|A|>|\mathbb N|$) and $B$ is countably infinite
    (i.e. $|B|=|\mathbb N|$), then $A-B$ must be uncountable.}

  \begin{proof}
    
  \end{proof}

\item \emph{For sets $S$ and $T$, prove that if $|S-T|=|T-S|$, then
    $|S|=|T|$. Note that $S$ and $T$ may not be finite!}

  \begin{proof}
    
  \end{proof}

\item \emph{Demonstrate that $\mathbb R$ may be partitioned into
    collections with the following properties:}
  \begin{enumerate}
  \item \emph{a partition $\mathcal A$ such that $\mathcal A$ is uncountably
    infinite and each set in $\mathcal A$ is countably infinite.}

  \item \emph{a partition $\mathcal B$ such that $\mathcal B$ is countably
    infinite and each set in $\mathcal B$ is uncountably infinite.}

  \end{enumerate}
\end{enumerate}
\end{document}
