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\begin{document}
\begin{center}

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



\noindent 

\begin{tabular*}{6.0in}{l@{\extracolsep{\fill}}l}
$R_1$ is a relation on $\mathbb{Z}$		&$R_2$ is a relation from $A=\{1,2,3,4\}$ \\
defined by					&to $B=\{1,2,3\}$ defined by\\
$\{(a,b)| a+b\leq3\}$				&$\{(a,b)|\gcd(a,b)=1\}$\\
~\\
$R_3$ is a relation on $\mathbb{Z}$		&$R_4$ is a relation on $\mathbb{R}$\\
defined by							&defined by\\
$\{(a,b)|x=y^2\}$					&$\{(a,b)|a\cdot b\geq 0\}$

\end{tabular*}

\begin{enumerate}
\item For each of the relations $R_i$ defined above, list five ordered pairs that are in the relation.

$R_1$:
\vspace{.3in}

$R_2$:
\vspace{.3in}

$R_3$:
\vspace{.3in}

$R_4$:
\vspace{.3in}

\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*}


\newpage
\item Find an example of a relation on a set that is both symmetric and antisymmetric.
\vspace{1in}

\item Let $R$ be the relation on the set $\{1,2,3,4,5\}$ containing the ordered pairs:\\ $(1,1)$, $(1,2)$, $(1,3)$, $(2,3)$, $(2,4)$, $(3,1)$, $(3,4)$, $(3,5)$, $(4,2)$, $(4,5)$, $(5,1)$, $(5,2)$, and $(5,4)$.  

\begin{enumerate}
\item Find $R\circ R$
\vspace{1.5in}

\item Find $R^2$
\vspace{.5in}

\item Find $R^3$
\vspace{1.5in}
\end{enumerate}

\end{enumerate}
\end{document}
