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\newtheorem{thm}{Definition}

\begin{document}
\begin{center}

{\Huge{More Relations}}\\
\end{center}

\begin{tabular*}{6.0in}{l@{\extracolsep{\fill}}l}
$R_1$ is a relation on $A=\{1,2,3,4\}$	&$R_2$ is a relation on $A=\{a,b,c,d\}$\\
defined by						&defined with the directed graph below:\\
$\{(a,b)|\max(a,b)=b\}$				&\\
~\\
$R_3$ is a relation on the set of sets 	&\\
defined by						&\\
$\{(A,B)|A\subseteq B\}$				&\\
~\\
$R_4$ is a relation on $\mathbb{Z}$		&\\
defined by						&\\
$\{(a,b)|a\leq b\}$					&\\

\end{tabular*}


\vspace{-1.8in}
\hspace{3in}\includegraphics[height=3in]{relationa}	

\begin{enumerate}


\vspace{-1in}
\item For each of the relations $R_i$ above, determine if $R_i$ has the properties listed (if applicable).  If the relation does not have the indicated property, identify an ordered pair (or set of ordered pairs) that exhibits the failure.

\begin{tabular*}{6.0in}{l@{\extracolsep{\fill}}l|@{\extracolsep{\fill}}l|@{\extracolsep{\fill}}l|@{\extracolsep{\fill}}l}
	&$R_1$	&$R_2$	&$R_3$	&$R_4$\\
	\hline\\
	&&&&\\
reflexive  &&&&\\
	&&&&\\
	\hline\\
symmetric &&&&\\
	&&&&\\
	\hline\\
antisymmetric  &&&&\\
	&&&&\\
	\hline\\
transitive  &&&&\\
	&&&&\\
	\hline\\
\end{tabular*}

\item Identify any $R_i$ that are equivalence relations.  Identify the equivalence classes.
\vfill

\item Identify any $R_i$ that are posets.
\vfill

\item Identify any $R_i$ that are totally ordered.
\vfill




\end{enumerate}
\end{document}
