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\headers{Math375}{Homework 5}{Due: Sept 17, 2015}

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\textbf{Directions:} You may work to solve these problems in groups, but all written work must be your own.  \textbf{Show your work}; See ``Guidelines and advice'' on the course webpage for more information.

\begin{enumerate}

\item Let $A = \{1,2,3,4,5,6,7,8\}$ and let $B=\{\{1\}, \{2,3,4\}, (5,6), 7\}$.
\begin{enumerate}
	\item Determine $|A|$ and $|B|$.
	\item Determine $A\cap B$.
	\item True or False: $\{4,3,2\} \in A$
	\item True or False: $\{4,3,2\} \in B$
	\item True or False: $(6,5) \in A$
	\item True or False: $(6,5) \in B$	
\end{enumerate}

\item Let $[n] = \{1,2,3,\ldots,n\}$.  Describe a way to pair the subsets of $[n]$ of size $k$ with the subsets of size $n-k$.  What can you conclude about $\binom{n}{k}$ and $\binom{n}{n-k}$?  Hint: if you find this question confusing, generate data for small cases.  With $n=5$ and $k=2$, list the subsets of $\{1,2,3,4,5\}$ of size $2$ in one column and size $n-k$ or $3$ in a second column.  Try to find a natural way to pair them up.  If it still unclear, generate more data.  When ready, generalize.
\end{enumerate}

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