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\lhead{MATH 311-02}
\chead{Problem Set \#5}

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\rfoot{\bf due March 25}

\begin{document}
\begin{enumerate}
\item \textbf{(9 points)} \emph{The following questions will explore
    this slightly obscure relation property.}
  
  \begin{definition}
    A relation $R$ on a set $S$ is \emph{antireflexive} if and only
    if, for all $a\in S$, $(a,a)\notin R$; in other words, $aRa$ is
    false for all $a\in S$.
  \end{definition}

  \begin{enumerate}
  \item \textbf{(2 points)} \emph{Demonstrate, either by example or
      explanation, that there exist relations which are neither
      reflexive nor antireflexive.}
    \onehalfspacing



    \singlespacing
  \item \textbf{(7 points)} Prove that for a set $S$, if $R\subseteq
    S\times S$ is an antireflexive, symmetric, and transitive
    relation, then $R=\emptyset$.
    \onehalfspacing



    \singlespacing
  \end{enumerate}

\item \textbf{(25 points)} \emph{The following proofs concern unions
    and intersections of relations; if $R_1$ and $R_2$ are considered
    as subsets of $S$, we may take $R_1\cup R_2$ and $R_1\cap R_2$ to
    represent the ordinary operations on these sets.}
  \begin{enumerate}
  \item \textbf{(5 points)} \emph{Prove or disprove that, for
      relations $R_1$ and $R_2$ on $S$, if either $R_1$ or $R_2$ is
      reflexive, then the relation $R_1\cup R_2$ is reflexive.}
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  \item \textbf{(5 points)} \emph{Prove or disprove that, for
      relations $R5_1$ and $R_2$ on $S$, if both $R_1$ and $R_2$ are
      symmetric, then the relation $R_1\cup R_2$ is symmetric.}
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  \item \textbf{(5 points)} \emph{Prove or disprove that, for
      relations $R_1$ and $R_2$ on $S$, if both $R_1$ and $R_2$ are
      transitive, then the relation $R_1\cup R_2$ is transitive.}
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  \item \textbf{(5 points)} \emph{Prove that for equivalence relations
      $R_1$ and $R_2$ on $S$, $R_1\cap R_2$ is an equivalence relation
      \emph{(note: this is the intersection, whereas the previous
        questions discussed the union)}.}
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  \item \textbf{(5 points)} \emph{If $x\in S$ and $R_1$ and $R_2$ are
      equivalence relations of $S$, what is the relationship between
      the equivalence classes of $x$ with respect to $R_1$, $R_2$, and
      $R_1\cap R_2$?}
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  \end{enumerate}

\item \textbf{(6 points)} \emph{Prove or disprove and salvage if
    possible: for $[a],[b]\in\mathbb Z_n$ for a positive integer $n$,
    if $[a]\cdot [b]=[0]$, then either $[a]=[0]$ or $[b]=[0]$.}
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%\item \textbf{(4 point bonus)} \emph{Prove that for a positive integer
%    $n$, the perfect squares lie in at most
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\end{enumerate}

\end{document}