Concepts in Abstract Math

Note:

I wrote these notes during my position as Course Instructor for MAT246 in the summer of 2025 at the University of Toronto. They are based on my teaching of the several instances of the course, as well as other courses like MAT315 and MAT224.

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. More set theory exercises

4. Combinatorics

Enumerative combinatorics

1. Permutations and combinations
2. The pigeonhole principle
3. Counting principles

Graph theory

4. Graphs

5. Topology

1. Metric spaces
2. Compactness

6. Complex Numbers

1. Complex numbers

Additional resources

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)