\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}
