\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 4, 2013}}
\begin{document}
\emph{For a nonempty set $S$ and relation $R$ thereon, let the
  \emph{negation} of $R$ be the relation $\overline R$ described by
  either of the two following (equivalent) characterizations:}
\begin{itemize}
\item \emph{Considering relations as subsets of $S\times S$,
    $\overline R=(S\times S)-R$.}
\item \emph{For every pair $a,b\in S$, the statement $a\mathrel R b$
    is true if and only if $a\mathrel{\overline R}b$ is false.}
\end{itemize}
\begin{enumerate}
\item \emph{Prove that $\overline{\overline R}$ and $R$ are the same
    relation.}

  \begin{proof}
    
  \end{proof}

\item %Uncomment EXACTLY ONE of the following three instructions!
% \emph{Prove that if $R$ is reflexive, then $\overline R$ is reflexive.}
% \emph{Prove that if $R$ is reflexive, then $\overline R$ is non-reflexive.}
% \emph{Demonstrate that reflexivity of $R$ says nothing about $\overline R$, by giving two examples where $R$ is reflexive, but where the reflexivity of $\overline R$ is different among the two.}

\item %Uncomment EXACTLY ONE of the following three instructions!
% \emph{Prove that if $R$ is symmetric, then $\overline R$ is symmetric.}
% \emph{Prove that if $R$ is symmetric, then $\overline R$ is non-symmetric.}
% \emph{Demonstrate that symmetry of $R$ says nothing about $\overline R$, by giving two examples where $R$ is symmetric, but where the symmetry of $\overline R$ is different among the two.}

\item %Uncomment EXACTLY ONE of the following three instructions!
% \emph{Prove that if $R$ is transitive, then $\overline R$ is transitive.}
% \emph{Prove that if $R$ is transitive, then $\overline R$ is non-transitive.}
% \emph{Demonstrate that transitivity of $R$ says nothing about $\overline R$, by giving two examples where $R$ is transitive, but where the transitivity of $\overline R$ is different among the two.}
\end{enumerate}


\end{document}
