Homework 2

Due: Sep 10th, 2021

This homework will be on Chapter 2.

Please create a LaTeX document with the following code.

Note: This LaTeX code is not the same as in Homework 1. We’re gradually adding more to it. This time, we’ve added: “fancy” page headers, a “claim” setup, and shortcut commands for \(\mathbf{N}\) and \(\mathbf{Z}\). You’ll be able to type \bfN and \bfZ as shortcuts instead of \mathbf{N} and \mathbf{Z}.

\documentclass[12pt,oneside]{amsart}

\title{Math 287 Homework 2}
\author{your name}
\date{September 10, 2021} % the due date of the homework

\usepackage[T1]{fontenc}
\usepackage{amsmath,amsfonts,amssymb,amsthm}
\usepackage[letterpaper,margin=1.5in]{geometry}
\usepackage{fancyhdr}
\pagestyle{fancy}

% Extra space between lines
\linespread{2.4}

\theoremstyle{remark}
\newtheorem{exer}{Exercise}
\newtheorem{claim}{Claim}[exer]

\newcommand{\bfN}{\mathbf{N}}
\newcommand{\bfZ}{\mathbf{Z}}


\newenvironment{answer}{\bigskip\noindent\emph{Answer.}}{\hfill$\diamond$\newline}

\begin{document}
\maketitle




\end{document}

Copy that, and paste it into your LaTeX document. Change “your name” in the author line to be your name.

LaTeX: The \newcommand in LaTeX can be used to define shortcuts and other commands. In this homework template we’re using it for the shortcuts \bfN and \bfZ. Try defining your own shortcut commands!

Problems

This is a list of problems from the textbook, given by reference number. Your homework is to write a proof of the statement.

Note: In this class, to prove a statement, you are allowed to use any statement (proposition, axiom, definition, etc.) that is in the textbook before the statement you are trying to prove. You can use previous-numbered statements even if they weren’t proved in class or on homework, or if they were on the homework but you haven’t done that problem yet.

  1. Proposition 2.7(iv).

    To start, copy the following LaTeX code and paste it into your document, between \maketitle and \end{document}:

    \newpage
\begin{exer}
Proposition 2.7(iv). Let $m,n,p,q \in \bfZ$.
If $m < n$ and $p < 0$ then $np < mp$.
\end{exer}

\begin{proof}
-enter your proof here-
\end{proof}

  2. Proposition 2.12(iii).

    To help you get started, you can copy this LaTeX code:

    \newpage
\begin{exer}
Proposition 2.12(iii). For all $m,n,p \in \bfZ$,
if $p < 0$ and $mp < np$ then $n < m$.
\end{exer}

\begin{proof}
-enter your proof here-
\end{proof}

  3. Proposition 2.26. In this problem, the textbook gives a proof. Your homework is to rewrite the proof in more detail.

    Imagine a student in the class is confused by the proof. Rewrite the proof in a way that would make sense and be clear for a confused student.

  4. Project 2.28.

    In this project, you are supposed to

    1. Determine for which natural numbers \(k^2 - 3k \geq 4\). As part of your homework, give a clear statement of your answer.

    2. Prove your answer.

    To help you get started, you can copy this LaTeX code:

    \newpage
\begin{exer}
Project 2.28. Determine for which natural numbers $k^2 - 3k \geq 4$
and prove your answer.
\end{exer}

\begin{answer}
\begin{claim}
$k^2 - 3k \geq 4$ for (enter which values of k work)
\end{claim}

\begin{proof}
(enter your proof here)
\end{proof}
\end{answer}

    LaTeX: Observe, we can write squares (and other powers) using LaTeX code k^2, k^3, and so on. As an experiment, try writing k^m+1 and k^{m+1}. The curly braces {...} group together terms in LaTeX.

  5. The definition of “gcd” on page 22 of the textbook, in terms of smallest positive integer combination, might be unfamiliar. Soon we’ll look again at gcd and relate this definition to the more familiar versions. This homework problem will answer a different question: What does this have to do with induction and the Well Ordering Principle? Why did gcd show up in Chapter 2?

    1. As a thought experiment, let’s try defining “gcd of rational numbers” like this. Suppose \(a\) and \(b\) are rational numbers, take the set of elements \(ax+by\) where \(x\) and \(y\) are rational numbers, and \(ax+by > 0\). Can we define “\(\gcd(a,b)\)” to be the smallest element of that set? Why or why not? Explain.

    2. Why does gcd of integers work? How does the Well Ordering Principle help?

Additional assignments, required

These items are assigned. Please do them! But you do not have to turn in anything. These items will not be graded.

  • Read Chapter 3 and Chapter 4 of the textbook.

  • Visit Detexify. Try entering some symbols such as integral, partial derivative, or square root to see how to type them in LaTeX.

Instructions

Use LaTeX to create a PDF. Upload your PDF to Gradescope. If you don’t have LaTeX on your computer, you can use Overleaf. Don’t submit the LaTeX source, just the PDF. Email your instructor (that’s me) if you have questions or need help.

Please include your name and the homework number within the document. Some additional formatting instructions are in the syllabus. To summarize:

  • Use a new page (\newpage) for each problem.
  • State which question you are answering and the actual question. Then, start your answer in a new paragraph.

You are encouraged to work together on the homework!

Additional reading, optional

These readings are optional (not required).