\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{graphicx}
\graphicspath{{./}{figs/}{../figs}{../}}
\usepackage{tikz}

%% ------ 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 5 $|$ Due February 17 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 3190 $|$ Introduction to Proof $|$
    Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-4mm}

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

\begin{exercise} % HW 5, Problem 1
Rewrite each of the following sentences so it has either the form ``\emph{If $P$, then $Q$},'' or ``\emph{$P$ if and only if $Q$},'' without changing the meaning.
    \begin{enumerate}
    \item A group is abelian whenever all nonidentity elements have order $2$.
    \item Whenever a subset of $\R^n$ is compact, it is closed and bounded.
    \item In order for $X$ to be complete, it is necessary that every Cauchy sequence converges.
    \item A ring that is a finite integral domain is necessarily a field.
    \item Being orthogonal is a sufficient condition for a matrix to be invertible.
    \item Being prime is enough to ensure that a nonzero element in $\Z[\sqrt{-5}]$ is irreducible. 
    \item A sequence converges only if it is Cauchy. 
    \item For a series $\sum a_n$ to diverge, it suffices that $a_n\not\to 0$.
    \item A set is open provided its complement is closed. 
    \item Every Cauchy sequence in $X$ converges unless $X$ is not complete.
    \item A matrix is invertible precisely when its determinant is nonzero.
    \item Every finite subgroup of a field’s multiplicative group is cyclic.
    \item Without compactness of $X$, continuity of $f$ need not imply boundedness of $f$.
    \item A matrix does not have an orthogonal set of eigenvectors unless it is symmetric.
    \end{enumerate}
\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 5, Problem 2
  Translate the following English sentences into symbolic sentences with quantifiers. The universe $U$ for each is given in parentheses.
  \begin{enumerate}[label=(\alph*)]
    \item For every positive number $x$, there is a unique number $y$ such that $2^y=x$. \;($U=\R$)
    \item For every complex number, there is at least one complex number such that their product is a real number. \;($U=\C)$
  \end{enumerate}
\end{exercise}

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

\smallskip

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

\begin{exercise} % HW 5, Problem 3
  Which of the following are true? Briefly justify your answer for each. The universe $U$ for each is given in parentheses. 
  \begin{enumerate}[label=(\alph*)]
    \item $(\forall x)(x+x\geq x)$\; ($U=\R$)
    \item $(\forall x)(x+x\geq x)$\; ($U=\N$)
    \item $(\forall x)(\exists y)(x+y=0)$\; ($U=\R$)
    \item $(\exists x)(\forall y)(x+y=0)$\; ($U=\R$)
    \item $(\forall y)(\exists x)(\forall z)(xy=xz)$\; ($U=\R$)
  \end{enumerate}
\end{exercise}

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


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


\begin{exercise} % HW 5, Problem 4
    If $P(x)$ is an open sentence, then
    \begin{enumerate}[label=(\alph*)]
        \item $\neg(\forall x)P(x)$ is equivalent to $(\exists x)\big(\neg P(x)\big)$
        \item $\neg(\exists x)P(x)$ is equivalent to $(\forall x)\big(\neg P(x)\big)$.
    \end{enumerate}
    This was stated in class, and Part~(a) was proven. Prove Part~(b). Rather than include it as an exercise in your manuscript, add it below the existing proof of Part~(a).
\end{exercise}

\smallskip

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


\begin{exercise} % HW 5, Problem 5
Let $f(x)$ be a real-valued function. The following sentence is the definition of what it means for $f(x)$ to be continuous at $c\in\R$:
\[
(\forall\epsilon>0)(\exists \delta>0)\big(|x-c|<\delta\,\Rightarrow\,|f(x)-f(c)|<\epsilon\big).
\]
Write the negation of this sentence and simplify it so it does not contain the $\neg$ symbol.
\end{exercise}

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

\smallskip

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

\end{document}
