\documentclass[11pt]{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 =.5in]{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{enumitem} % 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}
\renewcommand{\thethm}{\textbf{\arabic{thm}}}

\title{Math 2001 \\ Definitions: review}

\begin{document}
\maketitle

\begin{defn}[Set equality]
If $A$ and $B$ are sets, then $A=B$ if $x\in A \implies x \in B$ and $x \in B \implies x \in A$.
\end{defn}

\begin{defn}[Subset]
If $A$ and $B$ are sets, then $A \subseteq B$ if $x \in A \implies x \in B$.
\end{defn}

\begin{defn}[Power set]
If $A$ is a set, then $\scr P(A) = \{x : x \subseteq A\}$.
\end{defn}

\begin{defn}[Union]
If $A$ and $B$ are sets, then $A\cup B = \{x : x \in A \text{ or } x \in B\}$.
\end{defn}

\begin{defn}[Finite and infinite unions]
If $\{A_i\}$ is a collection of sets indexed by $I$, then 
\begin{align*}
\bigcup_{i \in I} A_i = \{x : x \in A_i \text{ for some } i \in I\}.
\end{align*}
\end{defn}

\begin{defn}[Intersection]
If $A$ and $B$ are sets, then $A\cap B = \{x : x \in A \text{ and } x \in B\}$.
\end{defn}

\begin{defn}[Finite and infinite intersection]
If $\{A_i\}$ is a collection of sets indexed by $I$, then 
\begin{align*}
\bigcap_{i \in I} A_i = \{x : x \in A_i \text{ for every } i \in I\}.
\end{align*}
\end{defn}

\begin{defn}[Set difference]
If $A$ and $B$ are sets, then $A - B = \{x : x \in A \text{ and } x \notin B\}$.
\end{defn}

\begin{defn}[Complement]
If $A$ is a set, then $\overline A = \{x : x \notin A\}$.
\end{defn}

\begin{exer}
Fill out the right side of each block.
\end{exer}

\vspace{10mm}
\begin{multicols}{2}
\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Set equality}}
	(-3,0) node {$A=B$};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Subset}}
	(-3,0) node {$A\subseteq B$};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Power set}}
	(-3,0) node {$x \in \scr P(A) $};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Union}}
	(-3,0) node {$x \in A \cup B $};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{In/finite union}}
	(-3,0) node {$x \in \ds\bigcup_{i \in I} A_i$};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Intersection}}
	(-3,0) node {$x \in A \cap B $};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{In/finite intersection}}
	(-3,0) node {$x \in \ds\bigcap_{i \in I} A_i$};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Set difference}}
	(-3,0) node {$x \in A - B $};
\end{tikzpicture}

\begin{tikzpicture}[x=9mm]
\draw (-5,-1) rectangle (5,1)
	(0,0) node {$\iff$}
	(0,.7) node {\textbf{Complement}}
	(-3,0) node {$x \in \overline A$};
\end{tikzpicture}
\end{multicols}

\pagebreak

All of the proof that we have been writing recently have been of the form, ``prove that $A\subseteq B$." In order to prove that $A \subseteq B$, we show that $A$ and $B$ satisfy the definition of subset. Namely, we show that if $x \in A$, then $x \in B$. In particular, the body of the proof should start with the statement, ``suppose $x \in A$" (or ``if $x \in A$"). Then after a series of logical deductions, the proof ends once we conclude that $x \in B$.

\begin{exer}
Arrange the following statements to give an outline for a proof that $(A \cap B) - C \subseteq (A - C) \cap (B - C)$. Justify each statement by citing the appropriate definition.

\begin{minipage}{.55\textwidth}
\begin{align*}
\underline{\hspace{10mm}} &\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}
\end{align*}
\end{minipage}
\begin{minipage}{.4\textwidth}
\begin{enumerate}[label={\alph*.}]
\item $x \in A$ and $x \in B$ and $x \notin C$
\item $x \in A \cap B$ and $x \notin C$
\item $x \in (A\cap B) - C$
\item $x \in A-C$ and $x \in B - C$
\item $x \in (A-C) \cap (B-C)$
\end{enumerate}
\end{minipage}
\end{exer}

\begin{exer}
Since unions are `or' statements, proofs involving unions often break into multiple cases. Arrange the statements to prove that $(A - C) \cup (B-C) \subseteq (A \cup B) - C$. Justify each statement by citing the appropriate definition.

\begin{minipage}{.55\textwidth}
\begin{align*}
\underline{\hspace{10mm}} &\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
\text{Case 1: }
\underline{\hspace{10mm}} &\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
\text{Case 2: }
\underline{\hspace{10mm}} &\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}\\
&\implies \underline{\hspace{10mm}} \qquad \text{(by definition of \hspace{1.5in})}
\end{align*}
\end{minipage}
\begin{minipage}{.4\textwidth}
\begin{enumerate}[label={\alph*.}]
\item $x \in A-C$
\item $x \in B-C$
\item $x \in A \cup B$ and $x \notin C$
\item $x \in (A \cup B) - C$
\item $x \in B$ and $x \notin C$
\item $x \in A$ and $x \notin C$
\item $x \in A$ or $x \in B$, and $x \notin C$
\item $x \in A-C$ or $x \in B-C$
\item $x \in (A - C) \cup (B - C)$
\end{enumerate}
\end{minipage}
\end{exer}

\begin{exer}
In a similar fashion, sketch proofs for the following statements. In some cases, the justification for a step might not be a definition, but information that was given in the statement of the problem.
\begin{enumerate}[label={\alph*.}]
\item Prove that if $X \subseteq A \cap B$, then $X \subseteq A$ and $X \subseteq B$.
\item Prove that $\scr P(A \cap B) \subseteq \scr P(A) \cap \scr P(B)$.
\item Prove that if $\scr P(A) \subseteq \scr P(B)$, then $A \subseteq B$.
\end{enumerate}
\end{exer}

\vfill


\textbf{Upcoming deadlines:}
\begin{itemize}
\item Due Friday, Feb 19: first draft of Proof 4.
\item Due Monday, Feb 22: final draft of Proof 1 and second draft of proof 2.
\item Due Wednesday, Feb 24: second draft of proof 3.
\end{itemize}
\end{document}