\documentclass[12pt]{article}
\usepackage[sc]{mathpazo}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=-.1in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in
%%
\usepackage{fancyhdr,fancybox}
\usepackage{amssymb,amsmath,amsthm}
\usepackage{url}
\usepackage{hyperref}
\usepackage{latexsym}
\usepackage{enumitem} % Clashes with \usepackage{enumerate}
\usepackage{multicol}
\usepackage{mathrsfs}
\usepackage{graphicx}
\graphicspath{{./}{figs/}{../figs}{../}}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{arrows.meta}

%% ------ Custom solution environment ---------
\usepackage{comment}
\specialcomment{solution}
  {\par\noindent\textbf{Solution.}\ } 
  {\par}                              
%\excludecomment{solution}  % COMMENT/UNCOMMENT this to show/hide solutions

%%----Custom Exercise environment-----
\newcounter{exer}    %% Make a special counter just for exercises
%\numberwithin{exer}{section}
\theoremstyle{definition}
\newtheorem{exmp}[exer]{Problem}
\newenvironment{exercise}[1][]
{\begin{exmp}[#1]}
{\end{exmp}}

%%------------------------------------

\def\0{\mathbf{0}}
\def\<{\langle}
\def\>{\rangle}
\def\longto{\longrightarrow}
\newcommand{\C}{\mathbb{C}} %% Some people prefer \CC or \CCC or \bbC
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}
%% \def\Not[1]{\overline{#1}}  %% Uncomment this for an alternative
\newcommand{\Red}[1]{\textcolor{red}{#1}}
\newcommand{\Blue}[1]{\textcolor{blue}{#1}}
\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Trace}{tr}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}

%% Put these after \begin{solution}, as necessary
\newcommand\AIpolish{\textbf{(p)}\; }
\newcommand\AIminor{\textbf{(m)}\; }
\newcommand\AImajor{\textbf{(M)}\; }
\newcommand\AIpolishminor{\textbf{(p,m)}\; }
\newcommand\AIpolishmajor{\textbf{(p,M)}\; }

\renewcommand{\footrulewidth}{1pt}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
%%
\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 10 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

%%------------------------------------------------------------------------

\begin{exercise} % HW 8, Problem 1
    For each $r>0$, define the \emph{open interval of radius $r$} as the set
    \[
    I_r=(-r,r)\subseteq\R.
    \]
    Determine the following unions and intersections. Briefly justify each answer. 
\begin{multicols}{4}
\begin{enumerate}[label=(\alph*)]
    \item $\displaystyle\bigcup_{r>0}I_r$
    \item $\displaystyle\bigcap_{r>0}I_r$
    \item $\displaystyle\bigcup_{0<r<1}I_r$
    \item $\displaystyle\bigcap_{r>1}I_r$
\end{enumerate}
\end{multicols}
\noindent Then, repeat these, but for the \emph{closed interval of radius $r$}, defined as $\overline{I_r}=[-r,r]$.
\end{exercise}

%\begin{solution} 
%\begin{enumerate}[label=(\alph*)]
%    \item $\displaystyle\bigcup_{r>0}I_r$
%    \item $\displaystyle\bigcap_{r>0}I_r$
%    \item $\displaystyle\bigcup_{0<r<1}I_r$
%    \item $\displaystyle\bigcap_{r>1}I_r$
%\end{enumerate}
%%
%\begin{enumerate}[label=(\alph*)]
%    \item $\displaystyle\bigcup_{r>0}\overline{I_r}$
%    \item $\displaystyle\bigcap_{r>0}\overline{I_r}$
%    \item $\displaystyle\bigcup_{0<r<1}\overline{I_r}$
%    \item $\displaystyle\bigcap_{r>1}\overline{I_r}$
%\end{enumerate}
%\end{solution}

\medskip
    
%%------------------------------------------------------------------------

\begin{exercise} % HW 8, Problem 2
    For each $r>0$, define the open and closed disks of radius $r$ as the following subsets of $\R^2$:
    \[
    D_r=\big\{(x,y)\in\R^2:x^2+y^2<r^2\big\},\qquad 
    \overline{D_r}=\big\{(x,y)\in\R^2:x^2+y^2\leq r^2\big\}.
    \]
    Determine the following unions and intersections. Briefly justify each answer. 
\begin{multicols}{4}
\begin{enumerate}[label=(\alph*)]
    \item $\displaystyle\bigcup_{r>0}D_r$
    \item $\displaystyle\bigcap_{r>0}D_r$
    \item $\displaystyle\bigcup_{0<r<1}D_r$
    \item $\displaystyle\bigcap_{r>1}D_r$
\end{enumerate}
\end{multicols}
\noindent Then, repeat them for the closed disks. Many of your answers will be in terms of open or closed disks.     
\end{exercise}

%\begin{solution} 
%\begin{enumerate}[label=(\alph*)]
%    \item $\displaystyle\bigcup_{r>0}D_r$
%    \item $\displaystyle\bigcap_{r>0}D_r$
%    \item $\displaystyle\bigcup_{0<r<1}D_r$
%    \item $\displaystyle\bigcap_{r>1}D_r$
%\end{enumerate}
%%
%\begin{enumerate}[label=(\alph*)]
%    \item $\displaystyle\bigcup_{r>0}\overline{D_r}$
%    \item $\displaystyle\bigcap_{r>0}\overline{D_r}$
%    \item $\displaystyle\bigcup_{0<r<1}\overline{D_r}$
%    \item $\displaystyle\bigcap_{r>1}\overline{D_r}$
%\end{enumerate}
%\end{solution}

\medskip

%%------------------------------------------------------------------------

\begin{exercise} % HW 8, Problem 3
    Prove that
    \[
    \big\{n\in\Z:2\mid n\big\}\cap\{n\in\Z:9\mid n\big\}\subseteq  \big\{n\in\Z:6\mid n\big\}.
    \]
\end{exercise}

\medskip

%\begin{solution} 
    % To do.
%\end{solution}

%%------------------------------------------------------------------------

\begin{exercise}
Suppose $I\neq\emptyset$ and $I\subseteq J$. Prove the following containments:
\begin{multicols}{2}
\begin{enumerate}[label=(\alph*)]
    \item $\displaystyle\bigcup_{\alpha\in I}A_\alpha\subseteq\bigcup_{\alpha\in J}A_\alpha$
    \item $\displaystyle\bigcap_{\alpha\in I}A_\alpha\supseteq\bigcap_{\alpha\in J}A_\alpha$.
\end{enumerate}
\end{multicols}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}

\medskip
    
%%------------------------------------------------------------------------

\begin{exercise} % HW 8, Problem 5
Let $A$ and $B$ be sets.
\begin{enumerate}[label=(\alph*)]
    \item Show that $\mathscr{P}(A)\cup\mathscr{P}(B)\subseteq\mathscr{P}(A\cup B)$.
    \item State and prove a general condition on $A$ and $B$ for which equality holds in Part~(a).
\end{enumerate}
\end{exercise}

%\begin{solution} 
    % To do.
%\end{solution}
    

%%------------------------------------------------------------------------

\end{document}

