This free and open-source textbook was written by me (Dana C. Ernst) and is designed to be used with an inquiry-based learning (IBL) approach to an introduction to proof course. The source files are located on GitHub.

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?

- An Introduction to Proof via Inquiry-Based Learning (complete set of notes)
- Title Page and Front Matter
- Preface
- 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: Induction
- 4.1 Introduction to Induction
- 4.2 More on Induction
- 4.3 Complete Induction

- Chapter 5: Three Famous Theorems
- 5.1 The Fundamental Theorem of Arithmetic
- 5.2 The Irrationality of $\sqrt{2}$
- 5.3 The Infinitude of Primes

- Chapter 6: Relations
- 6.1 Relations
- 6.2 Equivalence Relations
- 6.3 Partitions
- 6.4 Modular Arithmetic

- Chapter 7: Functions
- 7.1 Introduction to Functions
- 7.2 Images and Inverse Images of Functions
- 7.2 Compositions and Inverse Functions

- 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

If you’ve found an error or have suggestions for improvements, please let me know.

- Setting the Stage (PDF)
- Living Proof
- The Secret to Raising Smart Kids | Carol Dweck (article on Scientific American)
- 7 Things Growth Mindset is Not (article on Turn Around for Children)
- Videos on Growth Mindset and Productive Failure
- Grit: the power of passion and perseverance | Angela Lee Duckworth (6:12 min)
- How to Become a Better Learner | Learner Lab (13:30 min)
- Learning Like a Jungle Tiger | Trevor Ragan (5:23 min, similar to previous video)
- Productive Failure/Struggle Skateboard Video 1 (1:46 min)
- Productive Failure/Struggle Skateboard Video 2 (0:40 min)
- Productive Failure/Struggle BMX Video (1:39 min)
- Michael Jordan Failure Commercial (0:32 min)
- Mindsets: Fixed Versus Growth (2:19 min)
- Growth Mindset Animation (3:50 min)
- KhanAcademy interview with Carol Dweck about growth mindset (3:06 min)
- Make Mistakes | Michael Starbird (2:11 min)
- Ira Glass on the Creative Process (1:54 min)

- The State of Being Stuck | Ben Orlin (blog post on Math with Bad Drawings)
- The Chinese Farmer | Alan Watts (2:26 min)
- Do Schools Kill Creativity? | Ken Robinson (19:22 min)
- The Learner Lab Podcast
- The Subtle Art of the Mathematical Conjecture (article on Quanta Magazine)
- Lessons from My Math Degree That Have Nothing to Do with Math (6 min read on Medium)
- Is Mathematics Invented or Discovered? | Roger Penrose (4:14 min)
- What is Mathematics? The Most Misunderstood Subject (short article by Dr. Robert H. Lewis, Professor of Mathematics, Fordham University)
- Pictures of board work that we ran out of time to discuss.
*Note:*You should not assume that the solutions/proofs posted below are valid or sufficient.

Below are links to the take-home exams. If you are interested in using LaTeX to type up your solutions (this is optional), contact me and I will send you a link to the source file of the exam.

- Exam 2 (PDF). (Due Friday, March 13)
- Exam 3 (PDF). (Submit via BbLearn. Due by midnight on Saturday, April 11)
- Exam 4 (PDF). (Please email me a PDF of your work. Due extended to 5pm on Tuesday, April 28)
- Final Exam (PDF). (Please email me a PDF of your work. Due by 9am on Friday, May 8)

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MAT 226: Discrete Math

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MAT 431: Analysis

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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.