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\title{Week 6 Recitation}
\author{Robert Woodward}

\begin{document}
\maketitle

\begin{itemize}
\item (1 min max) Go over quiz from last week
\item (2 min max) Go over homework from last week.
\item Questions about lecture / homework so far?

\item Questions from refresher sheet...
\begin{enumerate}
\item Number 7: ``Prove that $n(2n+1)(2n-1)$
is always divisible by 3.''
\end{enumerate}

\item
Find the power Set of $\{1,2,\{3,4,5\}\}$:

First, consider how many elements the power set has. If the set
has $n$ elements, there will be $2^n$ elements. Because we have
3 elements, there will be $2^3 = 8$ elements.

Now, we can list all 8 elements:
\begin{quote}
$\{\emptyset, \{1\}, \{2\},\{\{3,4,5\}\}, \\
\{1,2\}, \{1,\{3,4,5\}\}, \{2,\{3,4,5\}\}, \\
\{1,2,\{3,4,5\}\}\}$
\end{quote}

\emph{Note:} We use $\{\}$ and not $()$.


\item 
Find the Cartesian Product for $A \times B \times C$ for $A = \{1,2,3\}$, $B = \{c\}$, $C = \{\{1,2,3\},\{a,b,c\}\}$:

\begin{quote}
\{(1,c,\{1,2,3\}), (1,c,\{a,b,c\}),(2,c,\{1,2,3\}), (2,c,\{a,b,c\}),(3,c,\{1,2,3\}), (3,c,\{a,b,c\})\}
\end{quote}

\item (Rosen 2.2:17b) Show that if $A$, $B$, and $C$ are sets, then $\overline{A \cap B \cap C} = \bar A \cup \bar B \cup \bar C$

Because the problem does not say what method we must use to prove this, we can take advantage of membership tables:

% Table generated by Excel2LaTeX from sheet 'Sheet1'
\begin{tabular}{|ccccc|cccc|}
\hline
$A$   & $B$   & $C$   & $A \cap B \cap C$ & $\overline{A \cap B \cap C}$ & $\bar A$ & $\bar B$ & $\bar C$ & $(\bar A \cup \bar B \cup \bar C)$ \\
\hline
\hline
0     & 0     & 0     & 0     & 1     & 1     & 1     & 1     & 1 \\
0     & 0     & 1     & 0     & 1     & 1     & 1     & 0     & 1 \\
0     & 1     & 0     & 0     & 1     & 1     & 0     & 1     & 1 \\
0     & 1     & 1     & 0     & 1     & 1     & 0     & 0     & 1 \\
\hline
1     & 0     & 0     & 0     & 1     & 0     & 1     & 1     & 1 \\
1     & 0     & 1     & 0     & 1     & 0     & 1     & 0     & 1 \\
1     & 1     & 0     & 0     & 1     & 0     & 0     & 1     & 1 \\
1     & 1     & 1     & 1     & 0     & 0     & 0     & 0     & 0 \\
\hline
\end{tabular}

\item (Last 10 minutes) Quiz

\end{itemize}

\end{document}