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\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}
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\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}