\documentclass[12pt]{article} \usepackage[utf8]{inputenc} \usepackage[dvipsnames]{xcolor} \usepackage[margin=1in]{geometry} \usepackage{enumitem} \usepackage{amsmath, amssymb, amsthm, mathrsfs} % \usepackage{bbold} % for mathbb{1} \usepackage{dsfont} % for mathds{1} \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={108}, pdfpagemode=FullScreen, } % 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}} \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{\BF}{\mathbb{F}} % field % \newcommand{\kring}{\mathbf{k}} % commutative ring k (e.g. field) % \newcommand{\st}{ \ : \ } % such that \newcommand{\st}{ \ | \ } % such that \newcommand{\divides}{ \,\vert \,} % a divides b \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 \newcommand{\F}{\mathbb{F}} % random field \newcommand{\Isom}{\mathrm{Isom}} % isometries group \DeclareMathOperator{\coker}{coker} % cokernel \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{\op}{^{\mathrm{op}}} % opposite thing \DeclareMathOperator{\Mod}{\mathbf{Mod}} % module category \newcommand{\RMod}{\,_R\Mod} % left R mods \newcommand{\ModR}{\Mod_R} % right R mods \newcommand{\bimod}{\text{-bimod}} % bimodules cat \DeclareMathOperator{\Grp}{\mathbf{Grp}} % cat of groups \newcommand{\SC}{\mathscr{C}} % category C \newcommand{\cat}{\SC} % category $C$... \newcommand{\CC}{\mathcal{C}} % category C (Rotman) or chain complex \DeclareMathOperator{\Hom}{\mathrm{Hom}} % module category \DeclareMathOperator{\Ab}{\mathbf{Ab}} % cat of abelian groups \DeclareMathOperator{\Set}{\mathbf{Set}} % cat of sets \DeclareMathOperator{\Tr}{Tr} % trace \newcommand{\Mat}{\mathrm{Mat}} % matrices \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} % calligraphic categoriies \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Tor}{Tor} \newcommand{\Hred}{\widetilde{H}} % reduced homology \newcommand{\cone}{\mathrm{Cone}} % mapping cone \newcommand{\Cone}{\cone} % mapping cone \DeclareMathOperator{\im}{im} % TOPOLOGICAL \newcommand{\interior}[1]{\overset{\circ}{#1}} % interior of space \newcommand{\RP}{\mathbb{RP}} % real projective \newcommand{\CP}{\mathbb{CP}} % complex projective \newcommand{\boldid}{\mathds{1}} % identity bolded % 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}} % round matrix \newcommand{\bpmat}{\begin{pmatrix}} \newcommand{\epmat}{\end{pmatrix}} % square matrix \newcommand{\bbmat}{\begin{bmatrix}} \newcommand{\ebmat}{\end{bmatrix}} \newcommand{\mznote}[1]{{\color{Periwinkle} \footnotesize #1}} % notes to students \newcommand{\mz}[1]{{\color{magenta} #1}} % notes to self \title{MAT 108 PS01} \author{Melissa Zhang} \date{Due Monday, April 6, 2026 at 9:00 pm on Gradescope} \begin{document} \maketitle \section*{Instructions} The goal of this problem set is to develop a solid foundation in arguing and writing mathematics. While there are very few problems, half of your focus should be on how to write mathematics clearly, using the conventions and standards of this course. \begin{itemize} \item When you begin writing proofs, it should feel like you are putting together a puzzle. You have a list of known facts (either axioms or statements we've already proven from those axioms), and your job is to fit them together into a coherent argument proving a new statement. \item For this problem set, the propositions you are instructed to prove come from the your textbook. You are only allowed to use the axioms / statements appearing before the stated proposition in the proof. \item \textit{How much detail is needed?} For this problem set, your solutions should be at about the same level of detail as the proof of Proposition 1.6 in the book. \item You absolutely must write in full, connected English sentences. Whenever possible, do not start a sentence with mathematical symbols. Do not use symbols like $\implies$, $\forall$, $\exists$, $\therefore$, etc. \end{itemize} Either handwrite your typeset\footnote{We will discuss how to typeset mathematics using TeX later in this course.} your solutions, and submit your solutions to Gradescope by the due date and time. \begin{itemize} \item Your solution must be \emph{neat}. You \textbf{will} be graded on style, which includes mathematical style and professionalism. \item It is your responsibility to mark where your solutions for each problem begins on Gradescope so that the TA and reader can properly grade your solutions. \end{itemize} %%%%%%%%%%%%%%%%%%%% \exercise \bea \item Read ``Notes for the Student'' (pages xv-xvi). \item Read Chapter 1. Make note of the notations, sentence structure, and syntax used in mathematical arguments. Also note how previous axioms or results are referenced in later proofs. \ee %%%%%%%%%%%%%%%%%%%% \exercise Prove Proposition 1.14: \begin{proposition} For all $m \in \Z$, $m \cdot 0 = 0 = 0 \cdot m$. \end{proposition} \begin{reminder} To prove an equation holds, we start at one end and use a chain of known equalities to arrive at the other end. \end{reminder} %%%%%%%%%%%%%%%%%%%% \exercise \bea \item Prove Proposition 1.24: \begin{proposition} Let $x \in \Z$. If $x \cdot x = x$, then $x=0$ or $x=1$. \end{proposition} \item Prove Proposition 1.26: \begin{proposition} Let $m,n \in \Z$. If $m \cdot n= 0$, then $m=0$ or $n=0$. \end{proposition} \ee %%%%%%%%%%%%%%%%%%%% \exercise In this exercise, we first define divisibility using only the multiplication operation $\cdot$ on $\Z$: \begin{definition} Let $m, n \in \Z$. If there exists a $q \in \Z$ such that $m = n\cdot q$, then we say \emph{$m$ is divisible by $n$}, or equivalently, \emph{$n$ divides $m$}. We denote this relationship by $n \divides m$. \end{definition} Prove the following statements. \bea \item 0 is divisible by every integer. \item If $m$ is an integer not equal to 0, then $m$ is not divisible by $0$. \item Let $x \in \Z$. If $x$ has the property that for all $m \in \Z$, $mx = m$, then $x = 1$. \ee \begin{hint} In higher math courses, it is often beneficial to read the book sections in addition to going to lecture. In particular, both lecture and the book are pointedly useful for part (c) of the above exercise. \end{hint} \end{document}