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\documentclass{math215}

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\doclabel{Math 215: Homework 9}
\docauthor{Your name here}
\docdate{March 30, 2012}

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\begin{document}

\begin{proposition}{4.30} For all $k,m\in\Nats$, where $m\ge 2$, 
\[
f_{m+k} = f_{m-1} f_k + f_m f_{k+1}.
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}


An integer $n$ is {\bf odd} if there exists an integer $j$ such that $n=2j+1$.

\begin{proposition}{9.A} Every integer is either even or odd, and no integer is both.
\end{proposition}
\begin{pf} 
Your proof goes here.  Use only material from Chapter 2 or earlier in your proof.
\end{pf}

\begin{proposition}{6.5}
Assume we are given an equivalence relation on a set $A$.  For all $a_1,a_2\in A$, either $[a_1]=[a_2]$ or $[a_1]\cap [a_2]=\emptyset$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{6.6}[Partial]  Let $A$ be a set and let $\Pi$ be a partition of $A$.  We define $a\sim b$ if there exists $P\in \Pi$ 
  such that $a\in P$ and $b\in P$.  Then $\sim$ is an equivalence relation.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{project}{6.7}  For each of the following relations defined on $\Ints$,
  determine whether it is an equivalence relation.  If it is, determine its
  equivalence classes.
\begin{enumerate}
\item $x\sim y$ if $x<y$.
\item $x\sim y$ if $x\le y$.
\item $x\sim y$ if $|x|= |y|$.
\item $x\sim y$ if $x\neq y$.
\item $x\sim y$ if $xy>0$.
\item $x\sim y$ if $x\mid y$ or $y\mid x$.
\end{enumerate}
\end{project}


\begin{proposition}{6.17} Let $m\in\Ints$.  Then $m$ is even if and only if
  $m^2$ is even.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{6.25} If $a\equiv a'\pmod n$ and $b\equiv b' \pmod n$ then
$$
a+b \equiv a'+b' \pmod{n}
$$
and
$$
ab \equiv a'b' \pmod{n}.
$$
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{project}{6.27}  Study the set $n$ such that $\Ints_n$ satisfies the
  cancellation property (Axiom 1.5).  You should form a conjecture, and then
  prove it.  Start working on the problem now, but it won't be due until the next assignment.
\end{project}


\end{document}