Math 321: Introduction to Advanced Mathematics
Class Meets Tuesday/Thursday 10:30–11:45 virtually via Zoom

See here for the syllabus.

See here for information about how the Zoom class meetings are structured.

The meeting id and password for the Zoom meeting can be found on the Laulima site for the course. The Laulima site also has the reading assignments and occasional short videos to accompany each class meeting.
Course Outline
The content of this course can be broken into four themes. For the most part, the themes will be covered in the order below. But at times we will take an early look at a theme or revisit an earlier theme with knowledge from a later theme.

I. Logic. The definitions and rules of firstorder logic.

II. Proofs. Strategies for proving mathematical statements, how to write proofs.

III. The language of mathematics. Sets, relations, functions.

IV. The limitations of the mathematical method. Alternative definitions, axioms, and rules of logic. How do we determine which are “correct”?
Tentative Weekly Schedule

(Aug 25&27) Introduction and overview

(Sept 1&3) Truth tables and logical connectives

(Sept 8&10) Set operations, variables, and logic

(Sept 15&17) Quantifiers

(Sept 22&24) Introduction to proofs

(Sept 29&Oct 1) Proof strategies: negations and conditionals

(Oct 6&8) Proof strategies: quantifiers

(Oct 13&15) Midterm; The universality of the logical connectives

(Oct 20&22) Induction

(Oct 27&29) Proof strategies: conjunctions, disjunctions, and biconditionals

(Nov 5) Relations

(Nov 10&12) Orders and functions

(Nov 17&19) Equivalence relations

(Nov 24) Cardinality

(Dec 1&3) Infinite sets and Cantor’s theorem

(Dec 8&10) Alternative axioms and logical rules; Proofs and Refutations
Announcements

Here are the 128 slides about cardinality.

Here are the 123 slides about equinumerosity.

Here are the 121 slides about functions.

Here are the 1124 slides about equivalence relations.

Here are the 1119 slides about equivalence relations.

Here are the 1117 slides about order relations.

Here are the 1112 slides about relations.

Here is a pdf with some examples of how to write induction proofs.

Reminder: No class on Tuesday, November 3!

Here and here are the 1022 and 1027 slides about mathematical induction.

I wrote up a sample snippet of mathematical writing, to demonstrate how mathematicians organize their writing. It’s similar to what I asked you to do for homework 5, but simpler (since I asked some of you to revise and resubmit homework 5, so I didn’t want to spoil the answer). You can find the pdf here and the LaTeX source here.

Here are some resources on guidelines for writing mathematics:

The midterm for the class will be next week (week of October 12). It will have two components, equally weighted in your grade. One component will be a written, takehome exam. In this, I will ask you to write a few proofs. Your final submission should adhere to the standards of written mathematics. You should use complete sentences, write in paragraphs, clearly explain each step, and so on. (See also the guidelines linked in the announcement below.)
The other component will be an oral exam. We will meet for 5 to 10 minutes for me to ask you a few brief questions about the material, and for me to see how you think about them. This will take place over the two class meetings for the week. I’ve emailed you a signup sheet to schedule your time. The way we will do it is, we will meet at the usual zoom meeting for the class. While I meet with you individually for the oral exams, you will meet in groups to read each others’ proofs for homework 5, and give each other feedback.

Here are the 106 slides about modular arithmetic.

You can find the slides for the lectures about proof strategies: 924 here and 927 here and 101 here.

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. 
You can see the slides from the first class here.
Homework
Homework is due every Thursday by 11:59pm. For Homework 1 onward, please submit via the Laulima site for the course. Each assignment will cover the material from Tuesday of that week and Thursday of the previous week.

Homework 0 Due Thursday, Sept 3

Homework 1 Due Thursday, Sept 10

Homework 2 Due Thursday, Sept 17

Homework 3: Do the exercises from the 917 & 922 inclass worksheet. Due Thursday, Sept 24

Homework 4 Due Thursday, Oct 1

Homework 5 Due Thursday, Oct 8

No homework due Thursday, Oct 22

Homework 6 Due Thursday, Oct 29

Homework 7 Due Thursday,
Nov 5Delayed to Nov 12 
Homework 8 Due Tuesday, Nov 24

Homework 9 Due Sunday, Dec 6