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\documentclass{math215}

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\usepackage{times,txfonts}

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\doclabel{Math 215: Homework 4}
\docauthor{Your name here.}
\docdate{February 15, 2013}

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\input{math215extras.tex}

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\begin{document}

\begin{proposition}{2.33} Let $A$ be a nonempty subset of $Z$.
  Suppose for some $b\in\Ints$ that $b\le a$ for all $a\in A$.
  Then $A$ has a least element.
\end{proposition}
Hint: Proving the Well-Ordering Principle was hard work.  But proving this
proposition should not be.  Just reformulate it into a form where you
can apply the Well-Ordering Principle.

\begin{proposition}{2.E} Suppose $n,m\in\Ints$ and $n>m$.  Then $-n<-m$.
\end{proposition}
\begin{pf}
Your proof goes here.
\end{pf}

\begin{proposition}{2.G}
Suppose $n,m,p\in\Ints$, $n>m$, and $p<0$.  Then $pn<pm$.
\end{proposition}
\begin{pf}
Your proof goes here.  Be lazy!
\end{pf}

\begin{proposition}{2.10} The equation $x^2=-1$ has no solution in $\Ints$.
\end{proposition}
\begin{pf}
Your proof goes here.
\end{pf}

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

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

\end{document}