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\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{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}}
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\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 7 $|$ Due March 3 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

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

\begin{exercise} % HW 7, Problem 1
Rewrite each of the following sets by listing their elements between braces. If it is an infinite set, write out enough elements for the reader to see the pattern, then use ellipses.
\begin{multicols}{2}
\begin{enumerate}[label=(\alph*)]
    \item $\{5x+2:x\in\Z\}$
    \item $\{x\in\Z:-2\leq x<7\dots\}$
    \item $\{5a+2b:a,b\in\Z\}$
    \item $\big\{\frac{m}{n}\in\Q:\left|\frac{m}{n}\right|<1\text{ and }1\leq n\leq 4\big\}$
    \item $\big\{x\in\R\mid x^4=1\big\}$
    \item $\big\{x\in\C\mid x^4=1\big\}$
    \item $\{3n:n\in\Z\text{ and }|2n|<8\}$
    \item $\{x\in\R:\sin(\pi x)=0\}$
\end{enumerate}
\end{multicols}
\end{exercise}

%\begin{solution} 
%    \begin{multicols}{2}
%    \begin{enumerate}[label=(\alph*)]
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%        \item $\{\}$
%    \end{enumerate}
%    \end{multicols}
%\end{solution}

\smallskip

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

\begin{exercise} % HW 7, Problem 2
Rewrite each of the following sets in set-builder notation.
\begin{multicols}{2}
\begin{enumerate}[label=(\alph*)]
    \item $\{3,5,7,9,11,\dots\}$
    \item $\{-2,-1,0,1,2,3,4,5\}$
    \item $\{\dots,-\frac{3\pi}{2},\pi,-\frac{\pi}{2},0,\frac{\pi}{2},\frac{3\pi}{2},\dots\}$
    \item $\{\dots,-\frac{8}{27},-\frac{4}{9},-\frac{2}{3},1,\frac{2}{3},\frac{4}{9},\frac{8}{27},\dots\}$
    \item $\{\dots,\frac{1}{27},\frac{1}{9},\frac{1}{3},1,3,9,27,\dots\}$
    \item $\{0,4,16,36,64,100,\dots\}$
    \item $\{2,4,8,16,32,64,\dots\}$
    \item $\{2,11,101,1001,10001,\dots\}$
\end{enumerate}
\end{multicols}
\end{exercise}

%\begin{solution} 
%    \begin{multicols}{2}
%    \begin{enumerate}[label=(\alph*)]
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%        \item $\{:\}$
%    \end{enumerate}
%    \end{multicols}
%\end{solution}

\smallskip

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

\begin{exercise} % HW 7, Problem 3 
    For each for the following sets, use TikZ to sketch the points/arcs/regions in the $xy$-plane.
    \begin{multicols}{2}
    \begin{enumerate}[label=(\alph*)]
    \item $\big\{(x,y):x\in[1,3],\,y\in[-1,2]\big\}$
    \item $\big\{(x,y):x\in\Z,\,y\in\R\big\}$
    \item $\big\{(x,y):x^2+y^2=4\big\}$
    \item $\big\{(x,y):x^2+y^2\leq 4\big\}$
    \item $\{1,2,3\}\times\{1,4\}$
    \item $\{1,2,3\}\times [1,4]$.
    \end{enumerate}
    \end{multicols}
\end{exercise}

\begin{comment}
\begin{solution}
    \[
    \begin{tikzpicture}[scale=.4]
    \tikzstyle{v} = [circle, draw, fill=black,inner sep=0pt, minimum size=1mm]
    \tikzset{>= {Stealth[length=6pt]}}
    \def\h{0.37}
    %%
    \begin{scope}[shift={(0,0)}]
    % Draw a circle if radius 1 centered at (2,0), filled gray
    \filldraw[fill=gray!30, draw=black] (2,0) circle (1);
    %%
    % Draw a rectangle w/ 2 specified corners, fill it in wtih gray
    \filldraw[fill=gray!30, draw=black] (-1,2) rectangle (2,4);
    %%
    %% Draw a line between two points
    \draw[thin] (-3,-2) to (-3,3);
    %%
    %% Draw a few points
    \node[v] at (-2,1) {}; 
    \node[v] at (4,2) {};
    %%
    %% Draw the x-axis and tye y-axis
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(a)};
    %% Draw tick marks on the x-axis
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    %% Draw tick marks on the y-axis
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};

    \end{scope}
    %%
    \begin{scope}[shift={(13,0)}]
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(b)};
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};
    \end{scope}
    %%
    \begin{scope}[shift={(26,0)}]
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(c)};
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};
    \end{scope}
\end{tikzpicture}
\]

\[
    \begin{tikzpicture}[scale=.4]
    \tikzset{>= {Stealth[length=6pt]}}
    \def\h{0.37}
    %%
    \begin{scope}[shift={(0,0)}]
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(d)};
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};
    \end{scope}
    %%
    \begin{scope}[shift={(13,0)}]
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(e)};
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};
    \end{scope}
    %%
    \begin{scope}[shift={(26,0)}]
    \draw[<->] (-5,0) to (5,0);
    \draw[<->] (0,-3.25) to (0,5);
    \node at (-4.5,5.5) {(f)};
    \foreach \i in {-4,...,4} {    
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \foreach \i in {-3,...,4} {    
        \draw[thin] (-.1,\i) to (.1,\i);
        \draw[thin] (\i,-.1) to (\i,.1);
    }  
    \node at (5.4,0) {\small $x$};
    \node at (0,5.4) {\small $y$};
    \end{scope}
\end{tikzpicture}
\]
\end{solution}
\end{comment}

\smallskip

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

\begin{exercise} % HW 7, Problem 4 
Use elementary logic to show that $\neg(A\subsetneq B)$ is equivalent to
\[
(A\nsubseteq B) \Or (A=B).
\]
\end{exercise}

%\begin{solution} 
    %% Reminder to cite AI usage, if applicable. 
    % Put it immediately after \begin{solution} (on the same line)
    %\AIpolish
    %\AImajor
    %\AIminor
    %\AIpolishmajor
    %\AIpolishminor
    %
    % To do.
%\end{solution}

\smallskip

%%------------------------------------------------------------------------
\begin{exercise} % HW 7, Problem 5 
Using elementary logic, and the equivalence of $A=B$ and $(A\subseteq B)\And(B\subseteq A)$, show that $A\neq B$ is equivalent to 
\[
(\exists a)\big[\big((a\in A)\And (a\notin B)\big)\Or\big((a\in B)\And(a\notin A)\big)\big].
\]
\end{exercise}

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

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

\end{document}

