Note:

These are only brief summaries, not full class notes. Many crucial details are missing.

If you skip lectures, you will fall behind. Reading these is no substitute for attending class.

Concepts in Abstract Mathematics

Table of Contents

Some general advice on learning mathematics can be found here.

1. Foundations

1. Introduction
2. Elementary set theory
3. Relations
4. Functions
5. The naturals
6. Induction
7. Additional practice problems

2. The Integers

1. Construction of the integers
2. Divisibility
3. Greatest common divisor and least common multiple
4. The fundamental theorem of arithmetic
5. Modular arithmetic
6. Fields
7. Solving a non-linear congruence

3. Infinity

1. Construction of the rationals
2. Gentle cardinal arithmetic
3. Indexed families of sets
4. Aleph naught
5. The continuum
6. Advanced problems

June break & midterm

4. Combinatorics

1. Enumerative combinatorics
2. Graph theory

5. Topology

6. Complex Numbers

Additional resources

My course is self-contained. Everything you need to understand and solve any question whose answer constitutes a part of your grade can be answered with the material I cover in my lectures. Some of it is mentioned on this site.

Important: I do not follow any book, notes or content other than my own, but most topics are covered in many other sources. Students are nevertheless encouraged to consult additional resources like the following. Here are a few:

All topics

• Dana C. Ernst - Introduction to Proof via Inquiry-Based Learning Link
• Mitchel T. Keller, William T. Trotter - Applied Combinatorics Link

Foundations

• Halmos - Naive Set Theory

The Integers

• Burton - Elementary Number Theory

Topology

• Royden and Fitzpatrick - Real Analysis (Chapter 1.4)
• Davidson and Donsig - Real Analysis and Applications (Chapter 4)