        \problemdata       % Takes 5 *mandatory* arguments
        {binrel-stats}            % latex-friendly label for the prob.
        {relations} %topic(s) of the problem content.
        {}               % Source (if known)
        {F01 q1-3} %Usage (list ones you are aware of).
        {}        % Last revision info (author, date).

\begin{problem}
In this problem, let $R$ be a binary relation on a set,
$A$, and $S$ be a binary relation from $A$ to a set, $B$.  Indicate
whether each of the following statements is \textbf{True} or
\textbf{False}.  For the false ones, \textbf{\emph{describe a
counterexample}}.

\begin{problemparts}

\problempart If $R= R^*$, then $R$ is an equivalence relation.
\solution{False. Counterexample: $A \eqdef \nats, R \eqdef \le$}

\vspace{0.7in}

\problempart If $R$ is symmetric, then so is $(R^{-1})^*$. \solution{True}

\vspace{0.7in}

\problempart If $R$ is the graph of a bijection $f:A \to A$, then $R^{-1}$ is
the graph of $f^{-1}$. \solution{True}

\vspace{0.7in}

\problempart If $A = B$, then $S \compose R = R \compose
S$. \solution{False.  Counterexample: $A = B \eqdef \set{1,2,3}, R \eqdef
\set{(1,2)}, S \eqdef \set{(2,3)},$ so $S \compose R = \set{(1,3)}, R
\compose S = \emptyset$}

\vspace{0.7in}

\problempart If $R = R^{-1}$ and $R$ is nonempty, then $R$ is
reflexive. \solution{False.  Counterexample: $xRy \eqdef\ x\neq y$ as
long as $\card{A} \ge 2$.}

\end{problemparts}

\end{problem}
