 \documentclass[11pt]{article}

\usepackage{latexsym}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage[all]{xy}


\xyoption{dvips}

\newtheorem*{goal}{Goal}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}

\setlength{\evensidemargin}{1in}
\addtolength{\evensidemargin}{-1in}
\setlength{\oddsidemargin}{1.5in}
\addtolength{\oddsidemargin}{-1.5in}
\setlength{\topmargin}{.5in}
\addtolength{\topmargin}{-1.5in}

\setlength{\textwidth}{16cm}
\setlength{\textheight}{23cm}
\newcommand{\dd}{\displaystyle}
\newcommand{\rook}{\hspace{-.1cm}\amalg\hspace{-.15cm}\bar{}}

\pagestyle{empty}


\begin{document}

\section*{Math 2001:  PHW4}
\subsection*{Due: February 10, 2016}

\begin{enumerate}
\item From the book do:
\begin{itemize}
\item 2.10: 2, 6, 10
\item 5: 2, 4 
\item 6: 4, 6, 14
\end{itemize}
\item Identify whether each of the following statements is true or false.  If it is true, prove it.  If it is false, then provide a counterexample.
\begin{enumerate}
\item Let $A$, $B$, and $C$ be sets.  Then
$$(A\cap B)\cup C=A\cap(B\cup C).$$
\item If $a,b\in \ZZ_{\geq 1}$ and  both $\sqrt{a}$ and $\sqrt{b}$ are irrational, then $\sqrt{ab}$ is irrational.
\end{enumerate}
\item A point $(m,n)$ in $\RR^2$ is a \emph{lattice point} if both $m,n\in \ZZ$.   Prove that the number of lattice points inside any circle centered at the origin is a number of the form $4k+1$ for some integer $k$ (note that you don't have to say what $k$ is).

Hint: Split the set of lattice points into subsets, depending on the quadrants.
\end{enumerate}

\end{document}