Math 299: Introduction to Mathematical Proof

Assignment #14 – Tuesday, April 14, 2020
$\newcommand{\negative}[1]{{\vphantom{#1}}^-{#1}}$

Type up the following proofs in LaTeX.  Use the semiformal proof style we have been using, one statement per line, and a reason stated for each line that needs one, with optional comments where you feel they would help the reader.  You can do that by clicking on the Homework Template link below on this page. To hand in your document, download the pdf and put your pdf file in your Dropbox folder with the appropriate filename. All variables and constants are assumed to be real numbers unless stated otherwise.

You can use any theorems prior to the chapter on real numbers (keeping in mind that theorems about the natural numbers do not automatically extend to theorems about the reals). Other than that you can only use the axioms and definitions about the real numbers, and any theorem you prove first to use in a later proof (of course). But you cannot use any theorems in the problem sections in Chapter 9 in the Lecture Notes.

  • 0. A warm up  $\negative{1}\neq 1$.
  • 1. One is positive  $0<1$
  • 2. Double negative  For all real numbers $x$, $$\negative{\left(\negative{x}\right)}=x$$
  • 3. Alternate additive inverse  $\negative{x}=\negative{1}\cdot x$
  • 4. Signed products  $\negative{x}\cdot\negative{y}=x\cdot y$
  • 5. Alternate multiplicative inverse  $x^-=\frac{1}{x}$
  • 6. Trichotomy  For any real numbers $x,y$ exactly one of the following is true:
    1. $x\lt y$
    2. $x=y$
    3. $y\lt x$
  • 7. Product of negatives is positive  $\negative{x}\cdot\negative{y}=x\cdot y$
  • 8. Some reals are natural  Consider the sequence of real numbers, $(N_k)_{k=0}^\infty$, defined at the beginning of section 9.4 in the notes. All of the Peano axioms hold for $(N_k)_{k=0}^\infty$. Let’s check a few.
    1. Show Peano Axiom N3 holds for $(N_k)_{k=0}^\infty$.
    2. Show Peano Axiom M1 holds for $(N_k)_{k=0}^\infty$.
    3. Show Peano Axiom I holds for $(N_k)_{k=0}^\infty$.

 Math 299 Online

  • Math 299 Online Classroom
  • Our new home away from home. Click here to enter our new Zoom classroom.
  • Zoom help – Zoom for Scranton students, how to install, how to use, documentation.

 Course Handouts

Proof Software

  • Lurch – a math word processor that can check your proofs! I designed and wrote this software with former Scranton math major Nathan Carter specifically with this course in mind.
  • Toy Proofs – a “toy” proof system I developed to introduce students to the concept of formal proofs.
  • Circle Dot Game – describing the definitions behind the game.
  • Toy Proof Talk Slides – slides from a talk I gave on Toy Proofs at the 2009 Joint Mathematics Meetings

Mathematical Writing and Typesetting

Overleaf
  • Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
  • Homework Template – click this link to start a new assignment (chose ‘Clone this project’ after clicking on this link).
  • Sample Document – an example document using the assignment style above.

Mathematical Writing and Typesetting (cont.)

LaTeX
  • LaTeX Cheat Sheet – a quick reference that can be printed on one sheet of paper 
    (posted by winston at stdout.org)
  • Detexify – a quick way to look up the name of a math symbol in \LaTeX by drawing it by hand.
LaTeX for laptops
  • LyX – a free math word processor based on LaTeX
  • MikTeX – install this and TeXnicCenter for LaTeX on your Windows computer
  • TeXnicCenter – install this and MikTeX for LaTeX on your Windows computer
  • MacTeX – Install this for LaTeX on your Mac computer