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\documentclass{math215}

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% following to use Times as the default font insteand of
% TeX's default font of Computer Modern.
\usepackage{times,txfonts}

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% in the header.
\doclabel{Math 215: Homework 4}
\docauthor{Your name here}
\docdate{February 17, 2012}

% I've provided a file (math215extras.tex) with some commonly used extra 
% commands. If you've downloaded it, you can include it in your document
% by uncommenting the line below.  Feel free to make changes to that file.
\input{math215extras.tex}

%%%% Main document starts here.

\begin{document}

\begin{proposition}{2.7(iii)} Let $m,n,p,q\in\Ints$.  If
$0<m<n$ and $0< p\le q$ then $mp<nq$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.11} Let $m\in\Nats$ and $n\in\Ints$.
  If $mn\in\Nats$ then $n\in\Nats$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.12(i)} If $m,n\in\Ints$, then
$$
-m < -n
$$
if and only if
$$
m > n.
$$
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.12(iv)} Let $m$, $n$, and $p\in\Ints$.
If $m\le n$ and $0\le p$, then $mp\le np$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.20} For all $k\in\Nats$, $k\ge 1$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.A} The integer $1$ is not divisible by $2$. That is,
$2\not\mid 1$.
\end{proposition}
\begin{proof}
Your proof goes here.
\end{proof}

\begin{proposition}{2.B} Let $A=\{3x-1:x\in\Ints\}$ and
  let $B=\{3x+8:x\in\Ints\}$.  Then $A=B$.
\end{proposition}
\begin{proof}
Your proof goes here.  This proposition is admittedly not very interesting; 
I have assigned so you can practice showing that two sets are the same.
Follow the template from our proof in class of Proposition 2.13.
\end{proof}

\end{document}