We will not be using a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. Any errors in the notes are no one’s fault but my own. In this vein, if you think you see an error, please inform me, so that it can be remedied. I reserve the right to modify them as we go, but I will always inform you of any significant changes. The notes will be released incrementally. Each link below is to a PDF file.

- An Introduction to Proof via Inquiry-Based Learning (complete set of notes)
- Title Page and Front Matter
- Chapter 1: Introduction
- 1.1 What is This Course All About?
- 1.2 An Inquiry-Based Approach
- 1.3 Your Toolbox, Questions, and Observations
- 1.4 Rules of the Game
- 1.5 Structure of the Notes
- 1.6 Some Minimal Guidance

- Chapter 2: Mathematics and Logic
- 2.1 A Taste of Number Theory
- 2.2 Introduction to Logic
- 2.3 Negating Implications and Proof by Contradiction
- 2.4 Introduction to Quantification
- 2.5 More About Quantification

- Chapter 3: Set Theory and Topology
- 3.1 Sets
- 3.2 Power Sets and Paradoxes
- 3.3 Indexing Sets
- 3.4 Topology of $\mathbb{R}$

- Chapter 4: Three Famous Theorems
- 4.1 The Fundamental Theorem of Arithmetic
- 4.2 The Irrationality of $\sqrt{2}$
- 4.3 The Infinitude of Primes

- Chapter 5: Induction
- 5.1 Introduction to Induction
- 5.2 More on Induction
- 5.3 Complete Induction

- Chapter 6: Relations (currently only Sections 6.1-6.3)
- 6.1 Relations
- 6.2 Equivalence Relations
- 6.3 Partitions
- 6.4 Order Relations

- Chapter 7: Functions
- 7.1 Introduction to Functions
- 7.2 Compositions and Inverses

- Chapter 8: Cardinality
- 8.1 Introduction to Cardinality
- 8.2 Finite Sets
- 8.3 Infinite Sets
- 8.4 Countable Sets
- 8.5 Uncountable Sets

- Appendix A: Elements of Style for Proofs
- Appendix B: Fancy Mathematical Terms
- Appendix C: Definitions in Mathematics

I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

Below are links to each exam.

- Exam 1 (take-home). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Monday, February 20)
- Exam 2 (take-home). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Monday, April 3)
- Exam 3 (take-home). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Wednesday, May 3)
*Note:*The original version of the exam contained a typo on Theorem E.2. As a result, you can skip Problem 4. - Final Exam (take-home). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due by 1pm on Friday, May 12)

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

Website

928.523.6852

Twitter

Instagram

Facebook

Strava

GitHub

arXiv

ResearchGate

LinkedIn

Mendeley

Google Scholar

Impact Story

ORCID

MAT 226: Discrete Math

MAT 320: Foundations

MAT 431: Analysis

This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

The source code is on GitHub.

Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.