- Write up and hand in a combinatorial proof for each of the following. Write your proofs in LaTeX using the Homework Article style and put your pdf file in Dropbox as usual. You cannot use any algebra, arithmetic, substitution, or previous formulas derived in or out of class. Just 100% pure counting! Bonus points will be awarded for nontrivial figures, but they aren’t required. Put all of the problems in a single document with a title and abstract, and restate the theorem you are trying to prove along with it’s problem number. You can cite this web page and the Lecture Notes as a reference.
- 1. Thm 12.10 (counting over computing) $$\binom{6}{2}\cdot\binom{4}{3}=\binom{6}{3}\cdot\binom{3}{2}$$
- 2. Thm 12.13 (binomial complement) $$\binom{m+n}{m}=\binom{m+n}{n}$$
- 3. Thm 12.15 (choose vs. permute) $$(n)_k=\binom{n}{k}\cdot k!$$
- 4. Thm 12.17 (good things come in pairs) $$\binom{2n+2}{k} = \binom{2n}{k} + 2\cdot\binom{2n}{k-1} + \binom{2n}{k-2}$$
- 5. Thm 12.26 (combination recursion) $$\binom{n+1}{k+1}=\binom{n}{k}+\binom{n}{k+1}$$
Math 299: Introduction to Mathematical Proof
Assignment #25 – Tuesday, May 7, 2024
Course Handouts
Lecture Notes – the latest lecture notes from previous instances of our course. May be revised frequently.
- Course Syllabus – our course syllabus
Mathematical Writing and Typesetting
Overleaf
Homework Template (Article style)
Click this link to start a new assignment (then copy the project in your list of Overleaf projects to make your own copy from my read-only original).
- Sample Document – an example document using the homework assignment style above.
- Homework
Template (Handout style)
Click this link to start a new assignment that is formatted more like a typical LaTeX article (then copy the project in your list of Overleaf projects to make your own copy from my read-only original). - Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
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
Math 299 Online
- Math 299 Online Classroom Our home away from home if we can’t meet in person for some reason. Click here to enter our Zoom classroom.
- Zoom help – Zoom for Scranton students, how to install, how to use, documentation.
Lurch!!
-
Lurch is a math word processor that can check your proofs!
- Quick Start Guide
- Topics – each one opens a blank document containing its rules and those from the topics above it.
- Math 299 Lurch – opens a blank document with all of the rules from the topics listed above available.
- Examples – a few proofs done in Lurch using the topics defined above.
- Lurch Syntax – a quick reference showing how to type various math expressions in Lurch
- Lurch Desktop App – the original desktop version of Lurch. It is being replaced by the web version above, but still has some features that haven’t been duplicated.
Toy Proof Software
- 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
- Scrambling Ice Cream – a word problem based on a variant of the Toy Proof system, Scrambler!.
- Lurch App – 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.