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\documentclass{math215}

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\doclabel{Math 215: Homework 7}
\docauthor{Your name here}
\docdate{March 8, 2013}

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\begin{document}

\begin{proposition}{2.33} Let $A$ be a nonempty subset of $\Ints$.
  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}{HW7.A}
  Let $m,n,p\in\Ints$ and suppose $p>0$.
  If $mp\le np$  then $m\le n$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\begin{proposition}{5.4}
  Let $A$, $B$, $C$ be sets.  
\begin{itemize}
  \item[(i)] $A=A$.
  \item[(ii)] If $A=B$ then $B=A$.
  \item[(iii)] If $A=B$ and $B=C$ then $A=C$.
\end{itemize}
\end{proposition}
\begin{proof}
  Your proof goes here. Be lazy!
\end{proof}

\begin{project}{5.12 (partial)}
For each of the following double implications $P\iff Q$ determine which 
of the implications $P\implies Q$ or $Q\implies P$, if any, are true.
For the ones that are true, prove them.  For the ones are are not
true, provide a counterexample.
\item[(ii)] $C\subseteq A$ or $C\subseteq B$ $\iff$ $C\subseteq(A\cup B)$
\item[(iii)] $C\subseteq A$ and $C\subseteq B$ $\iff$ $C\subseteq(A\cap B)$
\end{project}


\begin{proposition}{5.15 (DeMorgan's Laws)} Given two subsets $A,B\subseteq X$,
\begin{itemize}
  \item[(i)] $(A\cap B)^c = A^c \cup B^c$
  \item[(ii)] $(A\cup B)^c = A^c \cap B^c$
\end{itemize}  
\end{proposition}
\begin{pf}
Your proof here. Be lazy! Look at Lemma 5.14a and Corollary 5.14b
proved in class for useful results.
\end{pf}

\begin{proposition}{5.20} Let $A$, $B$, and $C$ be sets.
  \begin{itemize}
    \item[(i)] $A\times (B\cup C) = (A\times B)\cup (A\times C)$
    \item[(ii)] $A\times (B\cap C) = (A\times B)\cap (A\times C)$
  \end{itemize}
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\end{document}