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\documentclass{math215}

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\doclabel{Math 215: Homework 7}
\docauthor{Your name here}
\docdate{March 9, 2012}

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\begin{document}

\noindent In class we proved the following.  You can use it; {\bf do not} reprove it.
\begin{proposition}{2.D}  Let $m,n,p\in\Ints$ and suppose $p>0$.
  Then $m<n$ if and only if $mp<np$.
\end{proposition}

\hrule\vskip12pt

\noindent Now prove, while being as lazy as possible (i.e. taking advantage
of the fact that we proved Propositions 2.D and 2.12) the following.

\begin{proposition}{2.E}
  Let $m,n,p\in\Ints$ and suppose $p<0$.
  Then $m<n$ if and only if $mp>np$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}


\begin{proposition}{2.F}
  Let $m,n,p\in\Ints$ and suppose $p>0$.
  Then $m\le n$ if and only if $mp\le np$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\begin{proposition}{2.G}
  Let $m,n,p\in\Ints$.
  Then $m< n$ if and only if $m+p< n+p$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\hrule\vskip 12pt

\noindent Be lazy when proving DeMorgan's Laws.  Look at Lemma 5.14a and Corollary 5.14b
proved in class for useful results.

\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.
\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}

\newpage
\begin{project}{5.21} Let $A$, $B$, $C$, and $D$ be sets.  
  For each of the following statements, determine if it is true or false.
  If it is true, prove it.  If it is false, give a counterexample.
\begin{itemize}
  \item[(i)] $(A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D)$.
  \item[(ii)] $(A\times B)\cap (C\times D)=(A\cap C)\times (B\cap D)$.
\end{itemize}
\end{project}
\begin{pf}
Your proof here.
\end{pf}


% \begin{project}{5.A}  
% Let $A$, $B$, and $C$ be sets.
% Find an interesting alternative way to write
% $A\cap (B\cup C)$
% and prove that your alternative is correct.  Note that
% $(B\cup C)\cap A$ is not interesting.
% \end{project}
% \begin{pf}
% Your proof here.
% \end{pf}

\begin{proposition}{4.5} For all $k\in\Ints_{\ge 0}$, $k!\in\Nats$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\begin{proposition}{4.7 (i)}  For all $k\in\Nats$, $5^{2k}-1$ is divisible by $24$.
\end{proposition}
\begin{pf}
Your proof here.
\end{pf}

\end{document}
