\documentclass[12pt]{amsart} % Document type

% There is a lengthy top section...
% Copy and paste this into a new .tex file.
% Begin typing at the bottom of this document.

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Various add-ons that allow for greater customization of the document. Not all are necessary.

%\usepackage[T1]{fontenc} % Output encoding (suggested when text contains accented letters)
%\usepackage[utf8]{inputenc} % Input encoding (important when writing in foreign languages)
\usepackage[margin =.75in]{geometry} % Set page dimensions
\usepackage{parskip} % Custom paragraph spacing
\usepackage{setspace} % Allow for double-spacing using \doublespace
\usepackage{multicol} % Allows multiple columns on a page
\usepackage{mathtools,amssymb,amsfonts,amsthm} % Variety of useful things for math
\usepackage{euscript,mathrsfs,bbm} % Some additional fonts
\usepackage{color} % Color
\usepackage[usenames,dvipsnames,svgnames,table]{xcolor} % Extra colors

\usepackage{comment} % Customizable comment environments
\usepackage{newverbs} % Customizable verbatim environments
\usepackage{enumerate} % Customizable enumerate environments
\usepackage{tabularx} % Greater flexibility with tables
\usepackage{multirow} % Table elements span multiple rows

\usepackage{graphicx} % For inserting images
\usepackage{wrapfig} % Wrap text around figures
\usepackage{url} % For inserting urls
\usepackage[linktoc=all,colorlinks=true,linkcolor=blue,urlcolor=Purple,citecolor=cyan]{hyperref} % Internal and external links
\usepackage[all]{hypcap} % Image captions (requires hyperref)
\usepackage{pdfpages} % Insert pdfs
\usepackage{tikz} % For drawing pictures
    \usetikzlibrary{petri}
    \usetikzlibrary{trees}
%\usepackage[all]{xy} % Matrix-like diagrams
\raggedbottom
\allowdisplaybreaks

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Custom comment environments and various editing macros
% Comments can be turned off using \excludecomment{ }

\specialcomment{lecturenotes}{}{}\excludecomment{lecturenotes}
\specialcomment{answers}{ \textsc{Answer: }}{}
\specialcomment{remarks}{

\setlength{\leftskip}{5mm}
\begingroup\color{blue}}
{\endgroup

\setlength{\leftskip}{0mm}
}
\specialcomment{inremarks}{\begingroup\color{blue}}{\endgroup}
\newcommand{\ctypo}[1]{\colorbox{yellow}{#1}\hspace{-1mm}}
\newcommand{\ctex}[1]{\colorbox{Gold}{#1}}
\newcommand{\period}{\colorbox{cyan!50}{.}}
\newcommand{\comma}{\colorbox{cyan!50}{,}}

\newverbcommand{\cverb}{\small\color{purple}}{}
\newverbcommand{\everb}{\small\color{blue}}{}

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Custom column types for tabularx

\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}
\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}
\newcolumntype{C}[1]{>{\hsize=#1\hsize\centering\arraybackslash}X}

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Custom spacing commands

\begingroup
    \makeatletter
    \@for\theoremstyle:=definition,remark,plain\do{%
        \expandafter\g@addto@macro\csname th@\theoremstyle\endcsname{%
            \addtolength\thm@preskip\parskip
            }%
        }
\endgroup

\newcommand{\sss}{\vspace{2 mm}\noindent}
\newcommand{\lss}{\vspace{5 mm}\noindent}
\newcommand{\lls}{\vspace{7 mm}\noindent}
\newcommand{\ds}{\displaystyle}
\newcommand{\HR}{\vspace{5pt}\hrule height 1pt \vspace{5pt}}
\newcommand{\atext}[1]{\text{\quad #1 \quad }}
%\newcommand\Tstrut{\rule{0pt}{2.6ex}}         % = `top' strut
%\newcommand\Bstrut{\rule[-1ex]{0pt}{0pt}}   % = `bottom' strut

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Shortcuts for commonly used symbols

\newcommand{\ca}{\mathcal}
\newcommand{\fr}{\mathfrak}
\newcommand{\eucal}{\EuScript}
\newcommand{\scr}{\mathscr}
\newcommand{\bbf}{\mathbb{F}}
\newcommand{\bbn}{\mathbb{N}}
\newcommand{\bbr}{\mathbb{R}}
\newcommand{\bbq}{\mathbb{Q}}
\newcommand{\bbc}{\mathbb{C}}
\newcommand{\bbz}{\mathbb{Z}}
\newcommand{\bbp}{\mathbb{P}}
\newcommand{\bfa}{\mathbf{a}}
\newcommand{\bfb}{\mathbf{b}}
\newcommand{\bfc}{\mathbf{c}}
\newcommand{\bfu}{\mathbf{u}}
\newcommand{\bfv}{\mathbf{v}}
\newcommand{\bfx}{\mathbf{x}}
\newcommand{\ring}{\ca O}

\newcommand{\inv}{^{-1}}
\newcommand{\into}{\hookrightarrow}
\newcommand{\onto}{\twoheadrightarrow}
\newcommand{\isom}{\overset{\sim}{\to}}
\renewcommand{\iff}{\Leftrightarrow}
\renewcommand{\implies}{\Rightarrow}
\renewcommand{\impliedby}{\Leftarrow}
\newcommand{\contradiction}{\Rightarrow\Leftarrow}

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Shortcuts for matrices and vectors

\newcommand{\bmat}[1]{\begin{bmatrix}#1\end{bmatrix}}
\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\vmat}[1]{\begin{vmatrix}#1\end{vmatrix}}
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\newcommand{\Floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\vect}[1]{\langle #1 \rangle}

%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%
%
% Custom function names

\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\ad}{\operatorname{ad}}
\newcommand{\disc}{\operatorname{disc}}
\newcommand{\domain}{\operatorname{domain}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\ind}{\operatorname{ind}}
\newcommand{\nul}{\operatorname{Nul}}
\newcommand{\nullity}{\operatorname{nullity}}
\newcommand{\ord}{\operatorname{ord}}
\newcommand{\proj}{\operatorname{proj}}
\newcommand{\rank}{\operatorname{rank}}
\newcommand{\red}{\operatorname{red}}
\newcommand{\bbref}{\operatorname{rref}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\trunc}{\operatorname{Trunc}}
\newcommand{\vspan}{\operatorname{span}}

%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%
%
% Environment styles

\newtheorem{thm}{Theorem}[section]
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem*{prop*}{Proposition}
\newtheorem*{cor*}{Corollary}
\newtheorem*{lem*}{Lemma}
\newtheorem*{thm*}{Theorem}

\theoremstyle{definition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{defn}[thm]{Definition}
\newtheorem*{defn*}{Definition}
\newtheorem{claim}[thm]{Claim}
\newtheorem{example}[thm]{Example}
\newtheorem*{conj*}{Conjecture}
\newtheorem*{rd}{Read from the text}
\newenvironment{answer}{\noindent\textsc{Answer:}}{\hfill $\square$}

\newcounter{homework}
\newtheorem{hw}[homework]{Homework}
\newtheorem*{hw*}{Homework}
\newcounter{exercise}[homework]
\newtheorem{exer}[exercise]{Exercise}
\renewcommand{\theexercise}{\arabic{exercise}}

\theoremstyle{remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem*{remark*}{Remark}
\newtheorem{notation}[thm]{Notation}
\newtheorem*{notation*}{Notation}

\newcommand{\truefalse}{\qquad\textbf{T}\qquad \textbf{F}\quad :\quad}

\title{Math 2001 \\ Set operations}

\begin{document}
\maketitle

\begin{exer}
Express each of the following sets in set builder notation.
\begin{enumerate}[\quad i.)]
\setlength{\itemsep}{3mm}
\item $A \cap B =$
\item $A \cup B =$
\item $A - B =$
\item $\overline A =$
\end{enumerate}
\end{exer}

\begin{exer}
Let $A = \{\text{NJ, MD, MA, ME, CO}\}$, $B = \{\text{MA, ME, CO, UT}\}$, and $U$ be the set of states in the US. Give an explicit answer to each of the following.
\begin{enumerate}[\quad i.)]
\setlength{\itemsep}{3mm}
\item $A \cap B =$
\item $A \cup B =$
\item $A - B =$
\item $|\overline A| =$
\end{enumerate}
\end{exer}

\begin{exer}
Shade the region in the Venn diagram that corresponds to each of the following sets.

\begin{multicols}{4}
\begin{enumerate}[i.)]
\item $A \cap B$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $B \cup C$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $\overline A \cap C$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $A \cap B \cap C$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $(A \cap A) - (C \cup B)$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $(A \cap (A - C)) \cup B$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $A \cap ((A - C) \cup B)$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item $A \cap (A - (C \cup B))$

\begin{tikzpicture}
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}
\end{enumerate}
\end{multicols}
\end{exer}

\pagebreak

\begin{exer}
Give an expression that describes each shaded region.

\begin{multicols}{4}
\begin{enumerate}[i.)]
\item \

\begin{tikzpicture}
\fill[gray!50] (0:0) arc (180:240:1)
	arc (-60:-240:1)
	arc (180:0:1)
	arc (60:-120:1)
	arc (-60:0:1)
	arc (-60:0:1)
	arc (60:120:1)
	arc (60:120:1)
	arc (180:240:1);
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item \

\begin{tikzpicture}
\fill[gray!50] (-60:1) arc (-60:-240:1)
	arc (180:240:1)
	arc (180:120:1)
	arc (60:0:1)
	arc (-60:0:1)
	arc (60:-120:1);
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item \

\begin{tikzpicture}
\fill[gray!50] (-1.2,-1.2) rectangle (2.2,2.1);
\fill[white] (-60:1) arc (-60:-300:1)
	arc (120:-120:1)
	arc (-60:60:1)
	arc (120:240:1);
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}

\item \

\begin{tikzpicture}
\fill[gray!50] (-1.2,-1.2) rectangle (2.2,2.1);
\fill[white] (-60:1) arc (-60:-300:1)
	arc (120:60:1)
	arc (0:-60:1)
	arc (0:60:1)
	arc (120:180:1)
	arc (180:240:1);
\draw (-1.2,-1.2) rectangle (2.2,2.1)
	(-.9,1.8) node {$U$}
	(0,0) circle (1cm)
	+(210:.5) node {$B$}
	+(60:1) circle (1cm)
	++(60:1)+(90:.5) node {$A$}
	(1,0) circle (1cm)
	+(-30:.5) node {$C$};
\end{tikzpicture}
\end{enumerate}
\end{multicols}

\begin{enumerate}[i.)]
\setlength{\itemsep}{3mm}
\item
\item 
\item 
\item
\end{enumerate}
\end{exer}

\begin{exer}
Simplify each expression (if possible), then express the set in set builder notation.
\begin{enumerate}[i.)]
\setlength{\itemsep}{3mm}
\item $A \cup A \cup B$
\item $B-(B-A)$
\item $(B-B)-A$
\item $B - (C - A)$
\item $C - \overline A$
\item $\overline A \cap B \cap C$
\item $A \cup (B \cap \overline B)$
\item $\overline{A \cup B}$
\end{enumerate}
\end{exer}

\vfill

\begin{hw*}
Due Monday, February 1 at 6pm.
\begin{itemize}
\item Respond to the poll: \url{https://www.surveymonkey.com/r/GassertStudentVideoConsent}
\item Read Sections 1.5--7 from the text. 
\item Complete the following exercises (add these to your Overleaf file with the other book problems).
\begin{itemize}
\item Section 1.5: 4 (all).
\item Section 1.6: 2 (all).
\item Section 1.7: 7, 8. (You do not have to turn in the pictures.)
\end{itemize}
\end{itemize}
\end{hw*}

\end{document}

For all the definitions below, let $A$ and $B$ be sets.

\begin{defn*}
The \emph{union} of $A$ and $B$ is the set of elements that are either in $A$ or in $B$. 
\end{defn*}

\begin{notation*}
The symbol for union is $\cup$ (\cverb;\cup;) (think ``u" for ``union"):
\begin{align*}
A \cup B = \{x : x \in A \text{ or } x \in B\}.
\end{align*}
``The set $A$ union $B$ is the set of all $x$ such that $x$ is an element of $A$ or $x$ is an element of $B$."
\end{notation*}

When thinking of sets as boxes, the union of two sets is akin to taking the contents of the boxes $A$ and $B$ and dumping them into one big box, which we call $A \cup B$. In other words, $A \cup B$ contains everything that was in $A$ and everything that was in $B$.

\begin{defn*}
The \emph{intersection} of $A$ and $B$ is the set of elements that are simultaneously in both $A$ and in $B$. 
\end{defn*}

\begin{notation*}
The symbol for intersection is $\cap$ (\cverb;\cap;) (think ``n" for ``n-tersection"):
\begin{align*}
A \cap B = \{x : x \in A \text{ and } x \in B\}.
\end{align*}
``The set $A$ intersect $B$ is the set of all $x$ such that $x$ is an element of $A$ and $x$ is an element of $B$."
\end{notation*}

Going back to the box analogy, the intersection of two sets is like dumping $A$ and $B$ into a box, then picking out the duplicates. The duplicate items are the ones that were in both sets (one coming from $A$ and one coming from $B$).

\begin{defn*}
The \emph{difference} (or \emph{set difference}) of $A$ and $B$ is the set of all elements that are in $A$ but not in $B$. 
\end{defn*}

\begin{notation*}
The symbol for difference is $-$ (\cverb;-;, the minus sign):
\begin{align*}
A - B = \{x : x \in A \text{ and } x \notin B\}.
\end{align*}
``The set $A$ minus $B$ is the set of all $x$, where $x$ is an element of $A$ and $x$ is not an element of $B$."
\end{notation*}

\begin{defn*}
The \emph{complement} of $A$ is the set of all elements that are not in $A$.
\end{defn*}

\begin{notation*}
The symbol for complement is $\overline{A}$ (\cverb;\overline{A};). In set-builder notation:
\begin{align*}
\overline A = \{x : x \notin A\}.
\end{align*}
``The complement of $A$ is the set of all $x$, where $x$ is not an element of $A$."
\end{notation*}