%%% Template for MA 361 homework, after Eric J. Malm's hmcpset class
%
%%% PREAMBLE -----------------------------------------------------------
% Do make any changes before \begin{document}, except to add macros
%
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{boxedminipage}
\usepackage{graphicx}
\usepackage[margin=1in]{geometry}
\usepackage{mathpazo}
\usepackage{fancyhdr}

\setlength{\headheight}{15pt}
\setlength{\parindent}{0pt}

% include this to have lists enumerated by letters
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\roman{enumii}.}
\renewcommand{\labelenumiii}{\Alph{enumiii}.}



\newcommand{\collaborators}[1]{\begin{center}\large\sffamily{I collaborated with #1}\end{center}}

\newenvironment{problem}[1][]{%
  \bigskip% put space before problem statement box %
  \noindent\begin{boxedminipage}{\columnwidth}%
  \def\@tempa{#1}%
  \ifx\@tempa\empty\else%
    \textbf{#1}\hspace{0.5em}%
  \fi%
}{%
  \end{boxedminipage}%
}



\pagestyle{fancyplain}
\lhead{\textbf{\NAME}}
\chead{\textbf{\COURSE: HW\ \HWNUM}}
\rhead{\textbf{\DUEDATE}}
\cfoot{}


%%% MACROS -----------------------------------------------------------
% You may add more macros to this list
%
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}

\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Aut}{\operatorname{Aut}}




















\begin{document}
%%% NAME AND ASSIGNMENT NUMBER -----------------------------------------------------------
% Edit the values below to identify yourself and the assignment
%

\newcommand\NAME{Name:}  % your name
\newcommand\HWNUM{2}  % the homework number
\newcommand\DUEDATE{9/5/2019} % the due date
\newcommand\COURSE{MA 361}  % the course number

%%% COLLABORATORS -----------------------------------------------------------------------------------
% If you worked with other students on this assignment, acknowledge them here
% If you did not collaborate with anyone, comment out this line
%

%\collaborators{Ferdinand Georg Frobenius and Emmy Noether}


%%% PROBLEMS AND SOLUTIONS -------------------------------------------------------------------
% Include all problems statements
% For solutions, use the \begin{proof} environment
% Note that \begin{proof}[Solution] will replace the label "Proof" with "Solution", etc.
%

\begin{problem}[1]
Let $\ZZ$ be the set of all integers, and let
\[ 4\ZZ = \{ n \in \ZZ \mid \text{$n = 4k$ for some $k \in \ZZ$} \} \]
be the set of integers which are multiples of $4$.
\begin{enumerate}
\item Let $\sim_4$ be the relation on $\ZZ$ defined by
\[ a \sim_4 b \quad \text{if and only if} \quad a - b \in 4\ZZ. \]
Prove that $\sim_4$ is an equivalence relation by showing that it is \textbf{reflexive}, \textbf{symmetric}, and \textbf{transitive}.
\item How many equivalence classes does the relation $\sim_4$ have? Describe them.
\end{enumerate}
\end{problem}

%\begin{proof}
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%\end{proof}


\newpage


\begin{problem}[2]
Let $f \colon X \to Y$ be a surjective map of sets. For each $y \in Y$, the \textbf{fiber of $f$ over $y$} is the set
\[ f^{-1}(y) = \{ x \in X \mid f(x) = y \}. \]
[\textsc{caution:} The fiber $f^{-1}(y)$ is a \emph{subset} of $X$.]

\begin{enumerate}
\item Consider the surjection $f \colon \RR \to \RR_{\geq 0}$ given by $f(x) = x^2$. For each $r \in \RR_{\geq 0}$, describe the fiber $f^{-1}(r)$.

\item Let $f \colon X \to Y$ be a surjective map of sets. Prove that the relation
\[ x_1 \sim x_2 \quad \text{if and only if} \quad f(x_1) = f(x_2) \]
is an equivalence relation on the set $X$. Prove that the equivalence classes are the fibers of $f$.
\end{enumerate}
\end{problem}

%\begin{proof}
%
%\end{proof}



\end{document}
