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\documentclass[12pt]{article}
 
\usepackage[margin=0.7in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{tikz}
\usepackage{substr}

\IfSubStringInString{\detokenize{solution}}{\jobname}{%
  % Solutions
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{%
  % Question sheets
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\pagestyle{fancy}
\lhead{MATH 311-01}
\chead{Problem Set \#9\answer{ solutions}}
\rhead{\answer{}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due April 6, 2017}}

\newtheorem{prop}{Proposition}

% The command below redefines \neg (the logical negation character)
% to be a big tilde, in line with the usage in the book. Comment it
% out if you prefer the more traditional negation symbol.
\renewcommand*{\neg}{\mathord{\sim}}

\begin{document}
\begin{enumerate}
\item \question{For the following investigations, consider relations
    $R$ and $R'$ on a set $S$ such that $R\subseteq R'\subseteq
    S\times S$.}

  \begin{enumerate}
  \item \question{If $R$ is reflexive, is it necessarily the case that
      $R'$ is reflexive? Prove your assertion if true; furnish a
      counterexample if not.}

    \answer{}

  \item \question{If $R$ is symmetric, is it necessarily the case that
      $R'$ is symmetric? Prove your assertion if true; furnish a
      counterexample if not.}

    \answer{}

  \item \question{If $R$ is transitive, is it necessarily the case that
      $R'$ is transitive? Prove your assertion if true; furnish a
      counterexample if not.}

    \answer{}
  \end{enumerate}

\item \question{Let $S=\{1,2,3,4\}$. Either by explicitly constructing
    the relation as a set of ordered pairs, or by specifying the
    circumstances under which two elements of $S$ are related, fully
    describe a relation satisfying each of the eight conditions below,
    and briefly explain why your relation satisfies the conditions.}
  \begin{enumerate}
  \item \question{A relation $R_1$ on $S$ which is non-reflexive,
      non-symmetric, and non-transitive.}

    \answer{}

  \item \question{A relation $R_2$ on $S$ which is non-reflexive,
      non-symmetric, and transitive.}

    \answer{}

  \item \question{A relation $R_3$ on $S$ which is non-reflexive,
      symmetric, and non-transitive.}

    \answer{}

  \item \question{A relation $R_4$ on $S$ which is non-reflexive,
      symmetric, and transitive.}

    \answer{}

  \item \question{A relation $R_5$ on $S$ which is reflexive,
      non-symmetric, and non-transitive.}

    \answer{}

  \item \question{A relation $R_6$ on $S$ which is reflexive,
      non-symmetric, and transitive.}

    \answer{}

  \item \question{A relation $R_7$ on $S$ which is reflexive,
      symmetric, and non-transitive.}

    \answer{}

  \item \question{A relation $R_8$ on $S$ which is reflexive,
      symmetric, and transitive.}

    \answer{}

  \end{enumerate}

\item \question{Let $R$ be a relation on $\mathbb N$ such that $m$
    relates to $n$ if $m$ and $n$ have the same number of factors. In
    other, more technical words, we may explicitly define $R$ as:
    \[R=\bigl\{(m,n)\in\mathbb N\times\mathbb N:|\{d\in\mathbb N:d\mid
    m\}|=|\{d\in\mathbb N:d\mid n\}|\bigr\}\]
  }
  \begin{enumerate}
  \item \question{Prove that $R$ is an equivalence relation.}

    \answer{
      \begin{proof}
        
      \end{proof}
    }

  \item \question{Describe two of the equivalence classes of $R$.}

    \answer{}
  \end{enumerate}
\end{enumerate}

\end{document}
