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\begin{document}

\begin{center}
{\Large{\color{red}Assignment 3}}

{\it Due Wednesday, March 7, 06 at 11:00 AM in Class}

\end{center}

{\bf Remarks:} 
I may not grade all assignments, and may not grade all questions/parts on the assignments I choose to grade. 
You're welcome to ask me for help. 
Show your work and explain every step. If you don't provide enough explanation, you may get no credit or partial 
credit. You should do the assignment by yourself.

\vspace{.5 in}

\begin{enumerate}

\item Determine if each of the following binary relations is reflexive, symmetric, antisymmetric, 
transitive, equivalence relation, partial order:

\begin{enumerate}

\item Let $R$ be the relation on $\Z$ defined by 

\begin{center}
$a \text{ } R \text{ } b$ if and only if $b-5 < 2a < b+5$.
\end{center}

\item Let $R$ be the relation on $\Z$ defined by 

\begin{center}
$a \text{ } R \text{ } b$ if and only if $a+b$ is even.
\end{center}

\end{enumerate}

\item Define the following relation on $R^2$:

\begin{center}
$(a,b)R(c,d) \text{ iff  } a^2+b^2=c^2+d^2$. 
\end{center}

Find $[(\alpha,\beta)]$. What does $[(\alpha,\beta)]$ represent in the 
Cartesian plane? 


\item Let $U$ be the unit disc in $\R^2$, $X=\{x+\frac{1}{5} \text{ }| 
\text{ } x \in Z\}$ and let $Y=\Z \cup \{\frac{1}{5} \}$. 
Also, let $G=\{ (x,y) \in X \times X \text{} | \text{ } |x|\leq 
\frac{4}{7} \}$, $A= U \cap (Y \times Y)$, $B=U \cap G$, and $C=U \cap (\Z 
\times \Z)$. 

\begin{enumerate}

\item List the elements of $A$.

\item List the elements of $B$.

\item Find $A \bigtriangleup C$.

\item Find $B \bigtriangleup C$.

\end{enumerate}

\item Let 

\begin{center}

$A=\{(-1,2), (4,5), (0,0), (6,-5), (5,1), (4,3) \}$ 


$B=\{b \text{ }| \text{ } b= k^2 \text{ for some } k \in \Z \text{ and } 
(a,b) \in A \text{ for some } a \}$.

$C=\{x-4 \text{ }| \text{ } x \in Z \text{ and } \frac{x^2-17x+70}{-1576} 
\geq 0 \}$.

\end{center}

Find $(C \cup B) \cap \{ \{-3,1,2\}, \{1\}, 0, 5, 4, \{2\}, \{3\}, -2, -1, 
\phi\}$.

\item Let $\{u_n\}_{n=1}^{\infty}$ be defined by

\begin{center}
$u_n=\frac{2}{3}-(\frac{1}{6})(\frac{1}{4^{n-2}})$, $\forall n \in \N$.
\end{center}
 
 
Let $A=(-5,\frac{11}{3}] \cap [\frac{7}{4},5) \cap \Z^{+}$ and let 
$S=A-\{-4,7\}$. Find 
\[ \Sigma_{i \in S} \text{ } \Huge{ u_i}\].


\end{enumerate}

\end{document}