MATH 3040 is a course that is an introduction to mathematical proof. Emphasis is placed on clear, concise exposition and presentation in both written and verbal form. This class will follow an inquiry-based learning (IBL) format.

Textbooks

As a reference, we use The Art of Proof (TAP) by Matthias Beck and Ross Geoghegan. An electronic copy of the book is free to Cornell students, via the Cornell library. However, be aware that much of the material that you will be responsible for will not be directly from the textbook, and will only be available in class.

The attitude that we take towards any written material in this class—whether it is printed, on the board, or scribbled hastily on a napkin—is eloquently stated as follows:

"Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"

—Paul R. Halmos

A goal of this class is to gain practice in adopting this mindset and finding ways to successfully apply it, in short, to learn to "think like a mathematician," and in doing so, add this powerful tool to your intellectual toolbox.

Announcements

• Please see Canvas for the links to the rooms in which we will have classes and office hours.
• (03/09): On Tuesday, March 10 (tomorrow) at 5:30pm in the fifth floor lounge of Malott Hall, there will be an AWM talk on research in mathematics by Prof. Marie B.Langlois that should be accessible with the tools that we've been developing in this class. I encourage those of you who are interested to attend. (Plus, free tacos!)
• (02/25) Due to popular demand, Abby's office hours will henceforth be at 4:00-6:00pm on Wednesdays.
• For the week of January 27th, Abigail will have her office hours from 9:00-11:00am on Wednesday, as opposed to Thursday.
• (01/22): Here is the syllabus, the questionnaire and the student contract. If you were not present for the first class, please fill out the latter two and return them to me in person as soon as possible.
• The first class takes place on Wednesday, January 22.

Sheets

• Numbers
• Logic
• Induction
• Sets
• Functions and Cardinality
• Infinities
• Relations
• Graphs
• Groups
• Planes
• Surreals
• ...and Beyond

Class Assignments

Work on these assignments on your own and have them ready by the beginning of class on the date listed. These will be graded on a pass/fail basis.

• (03/09): Read [TAP], §13.1-3. Prepare solutions up to Proposition 7 on the Infinities sheet. Submit a proof of Theorem 3.
• (03/11): Read [TAP], §13.3-5. Prepare solutions up to Theorem 10 on the Infinities sheet. Submit a proof that the examples in Example 5 are countable (you may find the discussion in §13.5 to be useful).
• (03/13): Prepare (and revise) solutions to all questions on the Infinities sheet. Submit a proof of Proposition 7.

Here are some answers and comments for the previous class assignments.

• (04/06): Prepare solutions up to Question 7 on the Relations sheet and answer this poll.
• (04/08): Read [TAP], §6.1. Prepare solutions up to at least Question 10 on the Relations sheet. Do Question 13 or prepare an example of the correspondence for Theorem 13 (the reading should help). Answer the poll.
• (04/10): Prepare (and revise) solutions to all questions on the Relations sheet. Sign up for presentations.
• (04/13): Submit a proof of Theorem 12 from the Relations sheet, prepare solutions up to Proposition 8 on the Graphs sheet, and sign up for presentations.
• (04/15): Submit a proof of Proposition 8, prepare solutions up to Proposition 12, and sign up for presentations (and pose a question!).
• (04/17): Prepare (and revise) solutions to all questions on the Graphs sheet. Pick at least three questions from the class list, prepare general answers to them, and answer the poll.

Here are some answers and comments for the previous class assignments.

• (04/20): Read [TAP], Appendix D.1-2. Prepare solutions up to Proposition 5 on the Groups sheet, and sign up for presentations.
• (04/22): Prepare solutions for the remainder of the problems on the Groups sheet, and sign up for presentations.
• (04/24): Prepare your "cheat sheet" for Prelim II and take it on Canvas on Friday.
• (04/27): Prepare solutions up to Proposition 7 on the Affine and Projective Planes sheet, and answer the poll.
• (04/29): Prepare solutions up to Question 10 on the Planes sheet, submit your solution to Proposition 7, and sign up for presentations.
• (05/01): Prepare the remainder of the problems on the Planes sheet, and sign up for presentations.

Here are some answers and comments for the previous class assignments.

• (05/04): Prepare solutions up to Question 9 on the Surreals sheet, submit your solutions to Question 2 and Proposition 5, and sign up for presentations.
• (05/06): Read the solutions/comments below, prepare solutions up to Question 12 on the (updated!) Surreals sheet, submit your solution to Question 7, and answer the poll.
• (05/08): Read the (updated again!) Surreals sheet, prepare solutions up to Question 16, look at the class answers and questions and sign up for presentations.

Here are some solutions and comments for the Surreals sheet.

• (05/11): Prepare solutions to the questions on the ..and Beyond sheet, submit your answer to Question 4, and answer the poll.

Here are some answers and comments for the previous class assignments.

Weekly Homeworks

All these assignments should be submitted through Gradescope.

• Homework 1, due Friday, January 31. [template.tex]
• Homework 2, due Friday, February 7. [hw_template.tex], [groupwork_template.tex]. Solutions
• Homework 3, due Friday, February 14.
• Homework 4, due Friday, February 21.
• Homework 5, due Friday, February 28.
• Homework 6, due Friday, March 13. New group assignments
• Homework 7, due Friday, April 10. (Groupwork portion due next Friday.)
• Homework 8, due Friday, April 17.
• Homework 9, due Friday, May 1. New group assignments
• Homework 10, due Friday, May 8.

Topics

• Week 1 (01/22-01/24): Introduction to the course and its format, first proofs and presentations, ways to communicate a proof and why (TAP, pp. vii-xvii).
• Week 2 (01/27-01/31): Proofs with the integers (TAP, §1)
• Week 3 (02/03-02/07): Logic (TAP, §2.1, §3).
• Week 4 (02/10-02/14): Induction (TAP, §2.2-2.4, §4).
• Week 5 (02/16-02/21): Variations on induction and recursion (§4.5-6), Sets (§5).
• Week 6 (02/26-02/08): Functions and cardinality (§5.4, §9.1, §13.1).
• Week 7 (03/02-03/07): Review and revision, Prelim I. Solutions
• Week 8 (03/09-03/13): Countability, uncountability, and infinities (§13).
• Week 9 (04/06-04/10): Relations and Equivalence (§6).
• Week 10 (04/13-04/17): Graphs.
• Week 11 (04/20-04/24): Groups (§D), Prelim II. Solutions
• Week 12 (04/27-05/01): Affine and Projective Planes.
• Week 13 (05/04-05/08): Surreal Numbers.
• Week 14 (05/12): Last class.

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