\documentclass[12pt]{article}

\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{csquotes}

\usepackage{tikz}
\usetikzlibrary{positioning, scopes} 			%always use these
\usetikzlibrary{backgrounds}

%table spacing
\newcommand{\T}{\rule{0pt}{2.6ex}}
\newcommand{\B}{\rule[-1.2ex]{0pt}{0pt}}


\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\xor}{\operatorname{xor}}


\newtheorem{theorem}{Theorem}
\renewcommand*{\thetheorem}{I.\arabic{theorem}}
\newtheorem{exercise}[theorem]{Exercise}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}

\setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in}
\setlength{\marginparwidth}{0in} \setlength{\marginparsep}{0in}
\setlength{\topmargin}{0in} \setlength{\headheight}{0pt}
\setlength{\headsep}{0in} \setlength{\footskip}{.3in}
\setlength{\textwidth}{6.5in} \setlength{\textheight}{8.2in}%\setlength{\textheight}{9.2in}
\setlength{\parskip}{4pt}


\begin{document}

\begin{center}
{\large{\textsc{Homework 20}}} \\
\textsc{Math 3345 -- Autumn 2022 -- Kutler}
\end{center}

\thispagestyle{empty}

Please complete the following problems on your own paper. Solutions should be written clearly, legibly, and with appropriate style.

\begin{enumerate}
\item \textbf{{[Falkner Section 10 Exercise 11]}} Let $A$ and $B$ be sets. Show that $A \subseteq B$ if and only if $A \setminus B = \varnothing$.

\item \textbf{{[Falkner Section 10 Exercise 15 -- modified]}} Let $S$, $A$, and $B$ be sets.
\begin{enumerate}
\item Prove that $S \setminus (A \setminus B) = (S \setminus A) \cup (S \cap B)$.
\item Use part (a) to deduce that $A \setminus (A \setminus B) = A \cap B$.
\item Use part (a) to deduce that $B \setminus (A \setminus B) = B$
\end{enumerate}

\item \textbf{{[Falkner Section 10 Exercise 19 -- modified]}} Let $A$, $B$, and $X$ be sets.
\begin{enumerate}
\item Prove that if $A \subseteq B$, then $X \setminus B \subseteq X \setminus A$.
\item Prove that $A \subseteq X$ if and only if $A = X \setminus (X \setminus A)$. [\textsc{hint:} Use the previous problem to express $X \setminus (X \setminus A)$ in a simpler form.]
\item Suppose $A \subseteq X$. Prove that if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$.
\item Show, by giving an example, that the implication
\[ \text{if $X \setminus B \subseteq X \setminus A$, then $A \subseteq B$} \]
may be \textbf{false} if $A \not \subseteq X$.

That is, give an example of sets $A$, $B$, and $X$ such that $X \setminus B \subseteq X \setminus A$ and $A \not \subseteq B$.
\end{enumerate}
\end{enumerate}

\vspace{.05 in}

\begin{center}
\textbf{Practice Problems}
\end{center}

It is strongly recommended that you complete the following problems. There is no need to write up polished, final versions of your solutions (although you may find this a useful exercise). Please do not submit any work for these problems.

\begin{enumerate}
\item \textbf{{[Falkner Section 10 Exercise 3]}} Use set-builder notation to describe the sets
\[ A = \big\{ \{1\}, \{2\}, \{3\}, \ldots \big\} \]
and
\[ B = \big\{ \{1, 2, 3, \ldots \}, \{2, 4, 6, \ldots \}, \{3, 6, 9, \ldots\}, \ldots \big\}. \]
(Of course, you will need to make reasonable assumptions about the patterns in these examples.)

\item \textbf{{[Falkner Section 10 Exercise 5]}} Let $A$ be a set such that for each set $B$, we have $A \subseteq B$. Show that $A = \varnothing$.

\item \textbf{{[Falkner Section 10 Exercise 12]}} Prove Proposition 10.18(b): Let $A$ and $B$ be sets and let $x$ be any object. Then
\[ \text{$x \notin A \cap B$ if and only if $x \notin A$ or $x \notin B$.} \]

\item \textbf{{[Falkner Section 10 Exercise 24]}} Prove Proposition 10.34(b): Let $\mathscr{A}$ be a nonempty set of sets and let $X$ be any object. Then
\[ \text{$x \notin \bigg( \bigcap_{A \in \mathscr{A}} A \bigg)$ if and only if there exists $A \in \mathscr{A}$ such that $x \notin A$.} \]
\end{enumerate}




\end{document}
