\documentclass[11pt]{article} \usepackage[utf8]{inputenc} \usepackage[margin=1in]{geometry} \usepackage{enumitem} \usepackage{amsmath, amssymb, amsthm} \usepackage{xcolor} \usepackage{verbatim} % 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}}} % ENVIRONMENTS FOR CITING BOOK STATEMENTS \newcommand{\prop}[1]{\paragraph{Proposition #1}} \newcommand{\axm}[1]{\paragraph{Axiom #1}} \newcommand{\thm}[1]{\paragraph{Theorem #1}} % COMMANDS \newcommand{\solution}{{\color{blue} \textsc{\ \\ Solution.\ \ }}} \newcommand{\revisedsolution}{{\color{blue} \textsc{\ \\ Revised Solution.\ \ }}} \newcommand{\Z}{\mathbb{Z}} % the integers \newcommand{\N}{\mathbb{N}} % natural numbers \newcommand{\R}{\mathbb{R}} % real nubers \newcommand{\Q}{\mathbb{Q}} % rational numbers \newcommand{\divides}{\ \vert \ } \newcommand{\st}{ \ : \ } % such that \newcommand{\inv}{^{-1}} % inverse \newcommand{\exheading}[1]{\section*{Exercise #1}} \newcounter{exnum} % \setcounter{exnum}{1} % default 0 start \newcommand{\exercise}{ \stepcounter{exnum} \exheading{\theexnum} } %%%% \title{MAT 108: Problem Set 5} \author{(ADD NAME)} \date{Due 2/14/23 at 11:59 pm on Canvas} \begin{document} \maketitle \paragraph{Reminders:} \begin{itemize} %\item Put your name at the top! \item \textbf{No Monday 1-2 office hours with Dr.\ Zhang on 2/13.} TA office hours on 2/13 and Thursday, 2/16 office hours are as usual. \item Your homework submission must be typed up in full sentences, with proper mathematical formatting. Handwritten homework submissions will receive a score of 0. Solutions containing incomplete sentences or poor formatting will lose points. \item You will receive feedback on PS4 by next Tuesday, 2/14. PS4 revisions are due Friday, 2/17 at 11:59 pm. Underneath your old solution, type \begin{verbatim} \revisedsolution \end{verbatim} and then type your revised solution. \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise Consider the set $\Z/7\Z$, equipped with the operations addition ($+$) and multiplication ($\cdot$). We also have division, defined as a function \[ \text{division}: \Z/7\Z \times (\Z/7\Z - \{[0]\}) \to \Z/7\Z, \] which you will be able to define by $(x,y) \mapsto x y\inv$ after you've completed part (a) below. \begin{remark} For readability reasons, we will stop using the notation $[x]$ to represent the equivalence class of $x \in \Z$ under the ``mod 7" relation. In real life, mathematicians just write ``2" for $[2] \in \Z/7\Z$ if the context is clear. \end{remark} \begin{enumerate}[label={(\alph*)}] \item For each of the elements of $\Z/7\Z - \{0\}$, determine its (multiplicative) inverse. (Fill in the table provided.) \item Fill in the addition table provided below. \item Fill in the multiplication table provided below. \item An element $m \in \Z$ is called a \emph{square} if there exists some $n \in \Z$ such that $n\cdot n = m$. We can make the same definition in $\Z/7\Z$. Which elements of $\Z/7\Z - \{0\}$ are \emph{squares}? (Fill in the chart below.) \emph{($0$ is indeed a square, but it's extra special, so we consider it separately.)} \end{enumerate} \begin{remark} Just to be clear, this is an exploration problem. You don't need to write any proofs; just work out the calculations and fill in the tables below! \end{remark} \solution \renewcommand{\arraystretch}{1.5} \begin{enumerate}[label={(\alph*)}] \item % Inverse table: \begin{tabular}{|c|c|} \hline Element of $\Z/7\Z - \{0\}$ & Inverse element \\ \hline \hline 1 & \\\hline 2 & \\\hline 3 & \\\hline 4 & \\\hline 5 & \\ \hline 6 & \\ \hline \end{tabular} \item % Addition table: \begin{tabular}{|c|c c c c c c c |} \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & & & & & & & \\ 1 & & & & & & & \\ 2 & & & & & & & \\ 3 & & & & & & & \\ 4 & & & & & & & \\ 5 & & & & & & & \\ 6 & & & & & & & \\ \hline \end{tabular} \item % Multiplication table: \begin{tabular}{|c|c c c c c c c |} \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 0 & & & & & & & \\ 1 & & & & & & & \\ 2 & & & & & & & \\ 3 & & & & & & & \\ 4 & & & & & & & \\ 5 & & & & & & & \\ 6 & & & & & & & \\ \hline \end{tabular} \item % Squares: \begin{tabular}{|c|c|} \hline Squares: & \\ \hline Non-squares: & \\ \hline \end{tabular} \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise Let $x,y \in \R_{>0}$. Show that if $x < y$, then $0 < \frac{1}{y} < \frac{1}{x}$. \begin{remark} Remember that you have many properties and axioms to use from Sections 8.1 and 8.2. We didn't explicitly cover them in class if they come from axioms that $\Z$ also satisfied; in that case, the proofs for $\Z$ and for $\R$ are identical. \end{remark} \solution %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise In class, we talked about upper bounds (Section 8.4). In this problem, you will prove some analogous facts for lower bounds. \begin{enumerate}[label=(\alph*)] \item Let $B \subseteq \R$ be a nonempty subset. Give a precise definition for the \emph{infimum} $\inf(B)$ of $B$, i.e.\ the \emph{greatest lower bound} for $B$. \item Give a definition for $\min(B)$, the \emph{smallest element} of $B$. (Note that in class we only defined this function for subsets of $\Z$; your definition should be very similar.) Then, prove the following analogue to Proposition 8.49: \begin{proposition} Let $A \subseteq \R$ be nonempty. If $\inf(A) \in A$, then $\inf(A) = \min(A)$. Conversely, if $A$ has a smallest element, then $\min(A) = \inf(A)$ and $\inf(A) \in A$. \end{proposition} \end{enumerate} \solution \end{document}