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\begin{document} 

\noindent{\bf Math 214 \quad
Homework 7
\hfill Your name}
\date{}

\begin{enumerate}

\item (6 points) Solve the following problems in $\Z_n$. 
\begin{enumerate}
\item In $\Z_8$, express the following sums and products as $[r]$, where $0\le r<8$:  
 $$[3]+[6],  \quad  [3][6],  \quad [-13]+[138], \quad  
 [-13][138].$$

\item 
Let $[a], [b]\in \Z_8$. If $[a][b]=[0]$, does it follow that $[a]=[0]$ or $[b]=[0]$

\item Prove that for any prime $p$,  if $[a], [b]\in \Z_p$, then $[a][b]=[0]$ implies $[a]=[0]$ or $[b]=[0]$.
\end{enumerate} 

[Hint: $[ab] = [0]$ in $\Z_n$ means $ab$ is divisible by $n$.]

\item (a) (5 points) 
A relation $R$ is defined on $\Z$ 
by $(a, b)\in R$ if $|a-b|\le 2$.  
Which of the properties reflexive, symmetric, and transitive does the relation $R$ possess? Justify your answers.

(b) (5 points)
Let $R$ be a relation defined on $\Z-\{0\}$ by $(a,b)\in R$ if $ab>0$.  Show that $R$ is an equivalence relation on $\Z-\{0\}$. 

\item (8 points) Find relations on $S = \{1,2,3\}$ 
satisfying the following. Verify your answers.

(a) Reflexive, symmetric, not transitive.

(b) Reflexive, not symmetric, not transitive. 

(c) Symmetric, transitive,  not reflexive, 

(d) Symmetric, not reflexive, not transitive.


\item (8 points) 
Find relations on $\Z$
satisfying the following. Verify your answers.

(a) Reflexive, symmetric, not transitive.

(b) Reflexive, not symmetric, not transitive. 

(c) Symmetric, transitive,  not reflexive, 

(d) Symmetric, not reflexive, not transitive.

\item (a) (6 points)
Define the relation $R$ on $\R^2$ by $(x_1,y_1)R(x_2,y_2)$ if 
$|x_1|+|y_1| = |x_2|+|y_2|$. 

Prove that $R$ is an equivalence relation, and describe
the geometrical shape of the 

disjoint equivalence classes of $R$ in $\R^2$.

(b) (6 points) 
Consider the partition of $\R^2$ by straight lines $L_r = \{(x,y): x+y = r\}$
for each 

$r \in \R$. Show that $P = \{L_r: r \in \R\}$ is a partition of
$\R^2$, and
define an equivalence 

relation so that
$L_r$'s are the equivalence classes.

[Remark: We expect to see an answer saying that: $(x_1,y_1)R(x_2,y_2)$
if ....]


\item (6 points) Determine with explanation all the equivalence relations 
on $S = \{1,2,3\}$.

[Hint: Consider all partitions of $S$.]
\end{enumerate}

\end{document}


