Course format

Because of the pandemic, we will sometimes meet online, and we will also often meet outdoors (at locations Hiro will specify). Be sure to look out for e-mails with details.

If you are taking this class, but have any concerns about accessibility--including to fast-enough internet for online lectures, to technology (such as a laptop), et cetera--let Hiro know as soon as possible.

Class Meetings: MWF 11:00 AM - 11:50 AM.

Office Hours: Wed 1 PM - 3 PM.

Course description

This course will introduce you to tools that you will use repeatedly if you are a math major. These tools are not techniques like "derivatives" or "integrals;" they are tools of thought. How do you think about a problem? How do you know something is true? How do you convince someone something is true? We will develop these skills through examples provided by motivating some abstract ideas in mathematics--what is infinity, when are things equivalent for our purposes but not necessarily equal, how do we construct numbers, et cetera.

Beyond a fluency with the above topics, another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Textbook and resources.

Math CATS, sponsored by the Department of Mathematics, provides free drop-in and by-appointment math tutoring. Be warned that 3000-and-above levels of math courses often do not have ample supplies of tutors, so it will be very important for you to attend Office Hours and schedule study sessions with your friends/peers.

You do not need to buy a textbook for this course. The following are freely available resources:

1. Hammack's Book of Proof.
2. Ron Taylor's notes on introduction to proof.
3. Jacqueline A. Jensen-Vallin's notes.
4. Eric Baer's course at MIT.

The survey

Fill out this survey by Wednesday, August 25th, at 11:59 PM. It should take no more than 45 minutes. As part of the survey, I will ask you for your mathematical autobiography, so you should write one out and have it ready to be copy-pasted. Names will be removed from the survey responses, but all other results of the survey will be shared with the rest of the class.

The syllabus

Here is the course syllabus as of August 19.

Important dates

Exam I: Mon Oct 4 during class.
Final Exam: Mon Dec 6, 11:00 AM - 1:30 PM

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. However, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.

Recordings of Zoom Lectures

Can be found here.

Notes

Coming soon.

1. Mon, Aug 23. Sets.
Wed, Aug 25. (In-person) Discussion day to get started on Writing One.
2. Fri, Aug 27. Counting. (Notes have been updated to reflect class discussion!)
3. Mon, Aug 30. Bijections, and sizes of sets.
Wed, Sep 01. (In-person, optional virtual) Discussion day to get started on Writing Two.
Fri, Sep 03. (In-person, optional virtual) Discussion day to continue working on Writing Two, or to keep exploring Writing One.
Mon, Sep 06. No class, Labor Day.
4. Wed, Sep 08. (In-person, optional virtual) Subsets, power sets.
   Fri, Sep 10. (In-person, optional virtual) Discussion day to work on Writing Three.
5. Mon, Sep 13. (In-person; optional virtual) Bijections to power sets.
   Wed, Sep 15. (In-person, optional virtual) Discussion day to get started on Writing 04.
   Fri, Sep 17. (In-person, optional virtual) Discussion day to work on either of the topics from Monday and Wednesday.
   Mon, Sep 20. (Virtual) More groupwork on Writing 01.
6. Wed, Sep 22. (In-person, optional virtual) More on functions.
   Fri, Sep 24. (In-person, optional virtual) Discussion day.
   Mon, Sep 27. (In-person, optional virtual) Writing 01 solutions.
7. Wed, Sep 29. (In-person, optional virtual) Practice for exam on Monday, Oct 4. Solutions to practice problems.
   Fri, Oct 1. (In-person, optional virtual) Practice for exam on Monday, Oct 4 using the same problems from last class.
   Mon, Oct 4. Exam.
8. Wed, Oct 6. Set notation. Union and intersections. If time, getting started on Writing 07.
   Fri, Oct 8. Finishing up topics from last time.
9. Mon, Oct 11. Axiomatic Thinking. Peano's Axioms. What natural numbers are.
10. Wed, Oct 13. Induction.
11. Fri, Oct 15. Unique uniqueness of N.
12. Mon, Oct 18. Images, surjections, starting Cantor's Theorem.
    Wed, Oct 21. Cantor's Theorem.
    Fri, Oct 22. Finished proof of Cantor's Theorem.
13. Mon, Oct 25. Guest professor; in-class exercises and discussions.
    Wed, Oct 27. Guest professor; continue with exercises and discussions.
14. Fri, Oct 29. Inverses to bijections. (This lecture was not recorded, by accident.)
15. Mon, Nov 01. Infinitely many infinite sets. Continuum hypothesis.
16. Wed, Nov 03. Complements, products, function sets.In-person.
17. Fri, Nov 05. Having the same cardinality is transitive. Injection one way implies surjection the other way. Converses, logical equivalence.. In-person.
18. Mon, Nov 08. In-person.Cantor-Schroder-Bernstein Theorem.
19. Wed, Nov 10. Purely online. (Substitute professor.)Statements, truth tables, and/or/not, implications. Here are the Exercises for the class. There will be an online quiz as usual.
20. Fri, Nov 12. Purely online. (Substitute professor.)Implies, converse, contrapositive, for all, there exists. Here are the Exercises for the class. There will be an online quiz as usual.
21. Mon, Nov 15. Purely online. (Substitute professor.) Implications, contrapositives, converses.
22. Wed, Nov 17. In-person. Negation.
23. Fri, Nov 19. In-person. Well-ordering principle. Countability. Q is countable.
24. Mon, Nov 22. In-person. Equivalence relations.
    No class Wednesday (Thanksgiving break).
    No class Friday (Thanksgiving break).
25. Mon, Nov 29. Purely online. Equivalence relations, partitions, integers.
26. Wed, Dec 1. Purely online. Last day of class. R is uncountable. What's next?

Homework

Make sure to fill out the survey above by the first Wednesday of the semester, at 11:59 PM.
All homework assignments will be potentially shared with your classmates, so you may remove your names from your scanned/typed/written assignments. (When you upload your homework, I will know which assignments belong to whom, thanks to Canvas.)

1. Writing 1. Rational numbers. Deadline for Pre-Final Draft Submission: Friday, August 27, 11:59 PM.
   Extra Credit 1: Square roots. Deadline: Monday, August 30, 11:59 PM.
2. Writing 2. What's wrong? Deadline: Friday, September 3, 11:59 PM.
   Extra Credit 2. Light bulbs. Deadline: Monday, September 6, 11:59 PM.
   Writing 1 Second Draft. Deadline: Wednesday, September 8, 11:59 PM.
3. Writing 3. Thought I'd end up with Sean Deadline: Friday, September 10, 11:59 PM.
   Extra Credit 3. Composing bijections. Deadline: Monday, September 13, 11:59 PM.
   Optional: Writing 2 Second Draft (if you want to improve your grade on Writing 2). Deadline: Wednesday, September 15, 11:59 PM.
4. Writing 4. Bigger and bigger sets Deadline: Friday, September 17, 11:59 PM.
   Extra Credit 4. The natural number restaurant. Deadline: Monday, September 20, 11:59 PM.
   Writing 1 Third Draft Deadline: Wednesday, September 22, 11:59 PM.
5. Writing 5. Natural numbers and nextness Deadline: Friday, September 24, 11:59 PM.
   Extra Credit 5. Power sets Deadline: Monday, September 27, 11:59 PM.
   
6. Writing 6. Reviewing a topic Deadline: Friday, October 1, 11:59 PM.
   No Extra Credit this week.
   Writing 4 Second Draft Deadline: Wednesday, October 6, 11:59 PM.
   Writing 5, Second Draft. Deadline: Friday, October 8, 11:59 PM.
   Extra Credit 6. Sets union themselves. Deadline: Monday, October 11, 11:59 PM.
   Writing 1 Final Draft Deadline: Wednesday, October 13, 11:59 PM.
7. Writing 7. Unions and intersections Deadline: Friday, October 15, 11:59 PM.
   Extra Credit 7. Review. Deadline: Monday, October 18, 11:59 PM.
   
8. Writing 8. Different areas of knowledge Deadline: Friday, October 22, 11:59 PM.
   Writing 5 Final Draft Deadline: Wednesday, October 27, 11:59 PM.
   Writing 8.5. See Canvas. (Read lecture notes and ask three questions.) Deadline: Friday, October 29, 11:59 PM.
   Writing 4 Final Draft Deadline: Wednesday, November 3, 11:59 PM.
9. Writing 9. See Canvas. (Respond to peers on Canvas Discussion.) Deadline: Friday, November 12, 11:59 PM.
   
10. Extra Credit 10. (Note there was no extra credit 8 or 9.) Bijections between complements. Due Monday, November 15.
    Writing 10 (last one), first draft. Some concepts for calculus students Deadline: Monday, November 22, 11:59 PM.
    Writing 10 (last one), final draft. Some concepts for calculus students Deadline: Thursday, December 2, 11:59 PM. (No penalty for a late submission received by Friday, December 10, 11:59 PM)