\documentclass[12pt]{article}

\usepackage[margin=0.7in]{geometry}

\usepackage{fancyhdr,lastpage}

\usepackage{amsmath,amsfonts,amssymb,amsthm}
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{conj}{Conjecture}
\newtheorem{cor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}

\pagestyle{fancy}

\lhead{MATH 311-01}
\chead{Problem of the Day}
\rhead{YOUR NAME HERE}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{\textbf{due November 22, 2013}}
\begin{document}
\emph{Recall that for sets $A$ and $B$, the notation $A^B$ describes the set
of all functions from $B$ to $A$.}
\begin{enumerate}
\item \emph{Demonstrate, with an explicitly described function or pair
    of functions, that $|\{1,2,3,4\}^{\mathbb N}|=|\mathcal P(\mathbb
    N)|$.}

\item \emph{Demonstrate, with an explicitly described function or pair
    of functions, that $|\{1,2,3,4,5,6,7,8\}^{\mathbb N}|=|\mathcal
    P(\mathbb N)|$; generalize the idea you use here and above.}

\item \emph{Based on your previous result, prove that for any
    \emph{finite} set $S$ with at least two elements, $|S^{\mathbb
      N}|=|\mathcal P(\mathbb N)|$.}

\item \textbf{Bonus:} \emph{Prove that $|\mathbb N^{\mathbb
      N}|=|\mathcal P(\mathbb N)|$.}
\end{enumerate}
\end{document}
