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\documentclass{math215}

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\doclabel{Math 215: Homework 10}
\docauthor{Your name here}
\docdate{April 5, 2013}

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\begin{document}

\begin{proposition}{HW10.1} Let $A$ be a set, and let $\sim$
be an equivalence relation on $A$. Then the equivalence classes
of $\sim$ form a partition of $A$.
\end{proposition}
\begin{pf} 
\end{pf}

\begin{proposition}{HW10.2} Let $A$ and $B$ be sets.  Then
$$
(A\cup B) \setminus B \subseteq A.
$$
\end{proposition}
\begin{pf} 
\end{pf}

\begin{lemma}{6.13c}
Let $n\in\Nats$.  Suppose that $q$ and $r$ are integers
such that $0\le r \le n-1$ and
$$
qn+r =0.
$$
Then $q=0$ and $r=0$.
\end{lemma}
\begin{pf} 
\end{pf}

\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}
\end{proof}

\begin{lemma}{HW10.3} Suppose $n\in \Nats$, $a,b\in\Ints$, 
$2\le a\le n-1$, and $ab=n$.  Then
\[
2\le b\le n-1.
\]
\end{lemma}
\begin{proof}
\end{proof}

\begin{proposition}{6.28} Every integer greater than or equal to 2 can be
factored in to primes.
\end{proposition}
\begin{proof}
Wait until after Monday to start this one.
\end{proof}


\end{document}