Math 321: Introduction to Advanced Mathematics
Class Meets Mondays/Wednesdays/Fridays 10:30–11:20

See here for the syllabus.

Class meetings will be conducted over Zoom. Access info can be found on the Laulima site or the Discord for the class.

Outside of Zoom there are two main ways I will communicate with you: (1) This site, the public site for the class, where I will post homework, announcements, and any lecture notes; (2) a Discord channel, where I will also post announcements, where you can ask questions and discuss things with your classmates. Assignment submission will be done through Gradescope.
Course Outline
The main goal of this class is to teach you the basic methods and language of mathematics. In particular, the keystone of the class is the notion of proof, the means by which we justify mathematical statements as true. This class forms the foundation for upper division math classes, and what you learn here will serve you in further mathematical investigations.
Announcements

(4/15) Here is an article by logician Wilfrid Hodges relevant to this week’s Friday worksheet. He looks at a variety of “disproofs” of Cantor’s theorem which had been sent to him as editor of mathematics journal, analyzes where the mistakes are, and discusses some subtleties in logic and how teach mathematical argumentation which might underlie why some wrongly believe they’ve overturned Cantor’s theorem.

No class Friday, April 3rd on account of Good Friday

No class Friday, March 26th on account of Prince Kūhiō Day

(3/22) Here are solutions to the exam problems.

(3/5) The midterm is a takehome exam. It will be handed out next Monday (3/8) and be due the following Sunday (3/14). There will be no homework during midterm week.

(1/20) Here are some resources on guidelines for writing mathematics:

(1/20) In lecture, we discuss the fundamental theorem of arithmetic, stating that every positive integer has a unique prime factorization. You may think that this is an obvious fact which could not possibly be in need of a proof. If you think that, I suggest reading this blog post by Tim Gowers. If you don’t think it’s obvious, I’d suggest reading the post anyway. It is a nice little peak into how what we are talking about with the integers can be generalized, and why that generalization can be difficult.

No class Monday, January 18th an account of MLK Day

(1/13) For homework, you are strongly encouraged to use LaTeX. LaTeX is a program for typesetting mathematics, and is the standard in academic math, physics, and compsci. The learning curve can be a bit tough to start, but once you pick up the basics it can be much faster than mucking about with an equation editor. If you haven’t used LaTeX before, I suggest using Overleaf, a free online LaTeX editor. To help get you started, I made a template for homework.
The way LaTeX works is, you have a plaintext file which is complied into a pdf. (Overleaf does this part automatically for you.) The commands you put in the plaintext file then determine what shows up in the pdf. For example, to display the inequality $\sqrt[3]{x} \le \frac{x+1}{x^2+1}$ you would write
\sqrt[3]{x} \le \frac{x+1}{x^2+1}$
. The dollar signs say that what goes inside should be typeset as math, not as ordinary English. The various symbols =, +, 1, etc. are interpreted normally, while\frac
is the fraction command. The next two inputs, as inclosed in braces, are typset to be the numerator and denominator of the fraction. Similarly,\sqrt
is the square root command. It has an optional input, inclosed in square brackets. In this case, the optional input was used to make it a cube root.
Homework

Homework 1 Due Friday, January 22 Do the following exercises from page 8 of the textbook: 1.1, 1.3, 1.4, 1.9, 1.10.

Homework 2 Due Friday, January 29

Homework 3 Due Monday, February 8
Due Friday, February 5Here is a writeup of a solution for problem 4. (And here is the LaTeX source, in case it’s helpful to see.) 
Homework 4 Due Friday, February 19 Here is a writeup of a solution for problem 5. (And here is the LaTeX source, in case it’s helpful to see.)

Homework 5 Due Friday, February 26 Do the following exercises from pages 55–56 of the textbook: 5.1, 5.3, 5.9.

Homework 6 Due Friday, March 5

Homework 7 Due Monday, April 5 (Note the change in date!)

Homework 8 Due Friday, April 9

Homework 9 Due Friday, April 16

Homework 10 Due Friday, April 23
Inclass Worksheets

Week 4: No worksheet this week

Weeks 9–11: No worksheets these weeks
Lecture Notes

2/1, 2/3: Some logic See here for the filledout truth tables.

2/8, 2/10: Induction, I See here for a writeup of the proof that the least number principle and induction are equivalent.
Schedule
(Tentative, may change)

Week 1 (Jan 11) Intro to proofs: Chapter 1

Week 2 (Jan 18) More about proofs: Chapter 2

Week 3 (Jan 25) Number Theory: Chapter 3

Week 4 (Feb 1) Some Logic: No chapter

Week 5 (Feb 8) Mathematical Induction, I: Chapter 4

Week 6 (Feb 15) Mathematical Induction, II: Chapter 4

Week 7 (Feb 22) Discrete Mathematics: Chapter 5

Week 8 (Mar 1) Game Theory: Chapter 7

Week 9 (Mar 8) Pick’s Theorem: Chapter 8

Spring break!

Week 10 (Mar 22) Functions and Relations I: Chapter 11

Week 11 (Mar 29) Functions and Relations II: Chapter 11

Week 12 (Apr 5) Graph Theory: Chapter 12

Week 13 (Apr 12) Infinity I: Chapter 13

Week 14 (Apr 19) Infinity II: Chapter 13

Week 15 (Apr 26) Order Theory: Chapter 14

Week 16 (May 3) The Limits of Mathematics: No chapter