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\title{\sc Math 2100 / 2105 / 2350 -- Homework 9+10}
\author{due Thursday, \textbf{November 15}, at the beginning of class}
\date{}
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\begin{center} 
	\emph{This homework assignment was written in \LaTeX{}. You can find the source code on the course website.}
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\textbf{Instructions:} This assignment is due at the \emph{beginning} of class. \textbf{Staple your work} together (do not just fold over the corner). Please write the questions in the correct order. If I cannot read your handwriting, you won't receive credit. Explain all reasoning.

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{\bf \large Homework 9}

\begin{enumerate}
	\item Prove that there exists a positive integer $n$ such that $\frac{1}{n\ln(n)} < 0.0001$.
	\item Prove that any real number $r$ that makes the equation $r - \ds\frac{1}{r}=5$ true must be irrational. 
	\item Prove that if $a + b + c + d \geq 26$, then either $a \geq 3$, $b \geq 7$, $c \geq 7$, or $d \geq 9$.
	\item Use the pigeonhole principle to prove that given any five integers, there will be two that have a sum or difference divisible by $7$.
	\item Prove that at a completely full Milwaukee Bucks game at the new Fiserv Forum, there \emph{must} be at least two people that have both the same birthday \emph{and} the same first initial. (Note: you will have to look up the capacity of the new arena!)
	\item Show that if you pick 17 points from a square with side length $4$, then there must be 2 of those points that are within $\sqrt{2}$ of each other.
\end{enumerate}

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{\bf \large Homework 10}

\begin{enumerate}
	\item Prove or disprove: For any two sets $A$ and $B$, $\PP(A)\cup\PP(B) = \PP(A \cup B)$.
	\item Prove or disprove: For any two sets $A$ and $B$, $A \smallsetminus B = A \cap \overline{B}$.
	\item Prove the following set inequality:
	\[
		(\{n^2 - 1 : n \in \Z\} \cap \{2k : k \in \N\}) \subseteq \{4m:m\in\Z\}.
	\]
	\item Prove the following set inequality:
	\[
		(\{6k + 1 : k \in \Z\} \cup \{6m - 1: m \in \Z\}) \subseteq \{2n+1 : n \in \Z\}.
	\]
	\item Use induction to prove that for all $n \geq 1$, if $A$ is a set of size $n$, then the number of subsets of $A$ is $2^n$. (In other words, $|\PP(A)| = 2^{|A|}$.)
\end{enumerate}

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