\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}

\usepackage[margin=1in]{geometry}
\usepackage{enumitem}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{tikz, tikz-cd}
\usetikzlibrary{decorations.markings}
\usepackage{xcolor}
\usepackage{verbatim}
\usepackage{graphicx}
\usepackage{hyperref}
\hypersetup{
    colorlinks=true,
    linkcolor=blue,
    filecolor=magenta,      
    urlcolor=blue,
    pdftitle={Knot Theory},
    pdfpagemode=FullScreen,
    }
% \usepackage{ulem}

% THEOREM ENVIRONMENTS
\theoremstyle{definition}
\newtheorem*{theorem}{{Theorem}}
\newtheorem*{lemma}{{Lemma}}
\newtheorem*{proposition}{{Proposition}}
\newtheorem*{definition}{{Definition}}
\newtheorem*{corollary}{{Corollary}}
\newtheorem*{example}{{Example}}
\newtheorem*{remark}{{Remark}}
\newtheorem*{claim}{{Claim}}
\newtheorem*{question}{{Question}}

\theoremstyle{plain}
\newtheorem*{reminder}{{\textit{Reminder}}}
\newtheorem*{hint}{{\textit{Hint}}}



% COMMANDS
\newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}}
\newcommand{\Z}{\mathbb{Z}} % the integers
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\st}{ \ : \ } % such that
\newcommand{\inv}{^{-1}} % inverse
\newcommand{\into}{\hookrightarrow}  % injection
\newcommand{\onto}{\twoheadrightarrow} % surjection
\newcommand{\map}[1]{\xrightarrow{#1}} % named map
\newcommand{\id}{\mathrm{id}} % identity map
\newcommand{\Frac}{\mathrm{Frac}} % fraction field
\newcommand{\normal}{\unlhd} % normal subgroup of

\DeclareMathOperator{\img}{im} % image
\DeclareMathOperator{\sgn}{sgn} % sign of permutation
\DeclareMathOperator{\ord}{ord} % order of group element
\DeclareMathOperator{\lcm}{lcm} % least common multiple
% \DeclareMathOperator{\gcd}{gcd} % greatest common divisor

% \newcommand{\bpmat}{\begin{pmatrix}}
% \newcommand{\epmat}{\end{pmatrix}}


% knot theory
\newcommand{\lk}{\mathrm{lk}} % linking number


% SOME ENVIRONMENTS

\newcommand{\exheading}[1]{\section*{Exercise #1}}
\newcounter{exnum}
% \setcounter{exnum}{1} % default 0 start
\newcommand{\exercise}{
    \stepcounter{exnum} 
    \exheading{\theexnum}
    }

\newcommand{\bea}{\begin{enumerate}[label={(\alph*)}]}
\newcommand{\ee}{\end{enumerate}}




\title{MAT 150A HW04}
\author{[add your name here]}
\date{Due Tuesday, 10/31/23 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

\begin{reminder} 
Your homework submission \textbf{\large must be typed} (TeX'ed) up in full sentences, with proper mathematical formatting. The following resources may be useful as you learn to use TeX and Overleaf:
\begin{itemize}
    \item Overleaf's introduction to LaTeX: \\ \url{https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes}
    \item Detexify: \\ \url{https://detexify.kirelabs.org/classify.html}
\end{itemize}
\end{reminder}

\paragraph{Covered in this HW}
This homework is meant to be completed with knowledge of material from Lectures 1-12, with the purpose of preparing students to better understand upcoming material.


\paragraph{Grading} 

%Since Exam 1 is on Wednesday, October 25, all parts-of-problems in this homework will be graded out of 2 points. The goal is to practice and understand as many problems as you can. I advise against spending a disproportionate amount of time on any one problem; ask for help before the exam!

Some of the (parts of) problems will be graded in detail out of several points, and necessary feedback will be given. The rest will be graded out of 2 points. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

The purpose of this exercise is to strengthen your understanding of the term ``\textbf{well-defined}'' in mathematics.

\textit{Sometimes mathematicians
ask whether a function is
well defined. What they
mean is this: ``Does the rule
you propose really assign to
each element of the domain
one and only one value in
the codomain?''}

\hfill - \textit{The Art of Proof}, by Matthias Beck and Ross Geoghegan.

\bea

\item Prove that the following assignment is \textbf{not} a well-defined function between sets:
\begin{align*}
    \varphi: \Z/10\Z &\to \Z/7\Z \\
            \bar k &\mapsto \bar k.
\end{align*}
(Recall that $\bar k$ denotes the equivalence class of $k$ in $\Z/n\Z$.)

\item Prove that the following assignment \textbf{is} a well-defined function between sets:
\begin{align*}
    \varphi: \Z/10\Z &\to \Z/5\Z \\
            \bar k &\mapsto \bar k.
\end{align*}

\ee


\solution







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

The purpose of this exercise is to strengthen your understanding of the term ``\textbf{factors through}'' in mathematics.

\begin{definition}
Let $\varphi: G \to G'$ be a group homomorphism. 
If there exist 
\begin{itemize}
    \item a group $H$
    \item a homomorphism $\alpha: G \to H$ 
    \item a homomorphism $\beta: H \to G'$
\end{itemize}
such that $\varphi = \beta \circ \alpha$, then we say that $\varphi$ \textbf{factors through} $H$.  
\end{definition}

We usually capture this information in the form of a \textbf{commutative diagram}:
\begin{center}
\begin{tikzcd}
    G \arrow{r}{\varphi} \arrow[swap]{d}{\alpha} & G' \\
    H \arrow[swap]{ur}{\beta} & \\
\end{tikzcd}
\end{center}
Here, the descriptor ``commutative'' indicates that traveling along the horizontal arrow $\varphi$ is the same as traveling the length-2 path $\beta\circ \alpha$. 

\bea

\item Prove that the homomorphism 
$\varphi: \Z \to \Z/3\Z$, $k \mapsto \bar k$, factors through $\Z/9\Z$. 

\item  Prove that for all $\ell \in \N$, $\varphi$ factors through $\Z/3^\ell\Z$. 

\ee

\solution





\end{document}
