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\title{\sc Math 20 -- Homework 1}
\author{due Wednesday, April 8th}


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\textbf{Instructions:} Submit HW as a pdf on Canvas.  Write legibly to receive full credit.  Typed solutions are also acceptable.   
	
\begin{enumerate}[1.]

	\item How many different ways can the letters of the word \texttt{BOOKKEEPER} be rearranged? (For example, the letters in the word \texttt{EYE} can be rearranged in three ways: \texttt{EEY}, \texttt{EYE}, and \texttt{YEE}.)
	
	\item Suppose that you draw a five-card hand at random from a standard deck of $52$ cards. What is the probability that your hand contains two distinct pairs (for example: \texttt{3,3,Q,Q,K} or \texttt{6,6,9,9,K})?	
	
	\item Suppose that $A$ and $B$ are events. If $P(A \cup B) = 5/7$, $P(\overline{B}) = 4/5$, and $P(A) = 3/5$, then what is $P(A \cap B)$?

	\item \emph{(3 points)} Prove that if $A$ and $B$ are sets, then 
	\[
		\overline{A \cup B} = \overline{A} \cap \overline{B}
	\]
	and
	\[
		\overline{A \cap B} = \overline{A} \cup \overline{B}.
	\]
	\emph{Hint:} One way to prove that two sets $S$ and $T$ are equal is to first prove that $S \subseteq T$ and then prove that $T \subseteq S$. The only way both of these are true is if $S=T$.
	
	\item Give a proof by contrapositive of the following statement.
	\begin{center}
		\emph{Let $x$ and $y$ be integers.  If $x+y$ is even then either $x$ and $y$ are both even or $x$ and $y$ are both odd.}	
	\end{center}

	\item Give a proof by contradiction of the following statement.
	\begin{center}
		\emph{If $n$ is a positive natural number and $n^2$ is even, then $n$ is also even.}
	\end{center}
	
\end{enumerate}
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