2. Intro to Proof via IBL

An Introduction to Proof via Inquiry-Based Learning is a textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The reader is expected to construct their own understanding by engaging with the material. The content ranges over topics traditionally included in transitions courses: logic, set theory including cardinality, the topology of the real line, a bit of number theory, and more. The exposition guides and mentors the reader through an adventure in mathematical discovery, requiring them to solve problems, conjecture, experiment, explore, create, and communicate. Ultimately, this is really a book about productive struggle and learning how to learn.

What is This Book All About?

This book is intended to be used for a one-semester/quarter introduction to proof course (sometimes referred to as a transition to proof course). The purpose of this book is to introduce the reader to the process of constructing and writing formal and rigorous mathematical proofs. The intended audience is mathematics majors and minors. However, this book is also appropriate for anyone curious about mathematics and writing proofs. Most users of this book will have taken at least one semester of calculus, although other than some familiarity with a few standard functions in Chapter 8, content knowledge of calculus is not required. The book includes more content than one can expect to cover in a single semester/quarter. This allows the instructor/reader to pick and choose the sections that suit their needs and desires. Each chapter takes a focused approach to the included topics, but also includes many gentle exercises aimed at developing intuition.

Obtaining the Book

This book is unique. The book has been an open-source project since day one. The source and PDF versions of the book will always be available for FREE (see links below). In addition, the book has undergone the traditional review and editorial process with the American Mathematical Society/MAA Press and is available for purchase as a low-cost paperback. I am extremely grateful to the AMS/MAA Press for being willing to publish the book while maintaining the open-source license. I hope more textbooks can be published using the same model. Each year, I will donate any proceeds from the print version of the book to one or both of the Association for Women in Mathematics or the National Association of Mathematicians.

The first draft of the book was written in 2009. At that time, several of the sections were adaptations of course materials written by Matthew Jones (CSU Dominguez Hills) and Stan Yoshinobu (University of Toronto). The current version of the book is the result of many iterations that involved the addition of new material, retooling of existing sections, and feedback from instructors that have used the book. The current version of the book is a far cry from what it looked like in 2009.

• PDF of textbook compiled from the current source (Updated Updated October 31, 2024)
• PreTeXt/HMTL version of textbook compiled from the current source (Draft! Updated October 31, 2024)
• Low cost print version available via AMS/MAA Press
• PDF of textbook that matches the current print version by MAA Press
• Instructor Manual
• Errata (Updated Updated October 31, 2024)
• Source code available via GitHub
• Book review in MAA Focus
• Link to book at American Institute of Mathematics (AIM) Open Textbook Initiative

If you’ve found an error or have suggestions for improvements, please let me know by sending me an email or submitting an issue via GitHub. You can find the most up-to-date version of this textbook on GitHub. I would be thrilled if you used this textbook and improved it. If you make any modifications, you can either make a pull request on GitHub or submit the improvements via email. You are also welcome to fork the source and modify the text for your purposes as long as you maintain the Creative Commons Attribution-Share Alike 4.0 International License.

Why This Book?

Mathematics is not about calculations, but ideas. My goal as a teacher is to provide students with the opportunity to grapple with these ideas and to be immersed in the process of mathematical discovery. Repeatedly engaging in this process hones the mind and develops mental maturity marked by clear and rigorous thinking. Like music and art, mathematics provides an opportunity for enrichment, experiencing beauty, elegance, and aesthetic value. The medium of a painter is color and shape, whereas the medium of a mathematician is abstract thought. The creative aspect of mathematics is what captivates me and fuels my motivation to keep learning and exploring.

While the content we teach our students is important, it is not enough. An education must prepare individuals to ask and explore questions in contexts that do not yet exist and to be able to tackle problems they have never encountered. It is important that we put these issues front and center and place an explicit focus on students producing, rather than consuming, knowledge. If we truly want our students to be independent, inquisitive, and persistent, then we need to provide them with the means to acquire these skills. Their viability as a professional in the modern workforce depends on their ability to embrace this mindset.

When I started teaching, I mimicked the experiences I had as a student. Because it was all I knew, I lectured. By standard metrics, this seemed to work out just fine. Glowing student and peer evaluations, as well as reoccurring teaching awards, indicated that I was effectively doing my job. People consistently told me that I was an excellent teacher. However, two observations made me reconsider how well I was really doing. Namely, many of my students seemed to depend on me to be successful, and second, they retained only some of what I had taught them. In the words of Dylan Retsek:

“Things my students claim that I taught them masterfully, they don’t know.”

Inspired by a desire to address these concerns, I began transitioning away from direct instruction towards a more student-centered approach. The goals and philosophy behind inquiry-based learning (IBL) resonate deeply with my ideals, which is why I have embraced this paradigm. According to the Academy of Inquiry-Based Learning, IBL is a method of teaching that engages students in sense-making activities. Students are given tasks requiring them to solve problems, conjecture, experiment, explore, create, and communicate—all those wonderful skills and habits of mind that mathematicians engage in regularly. This book has IBL baked into its core.

This book is intended to be a task sequence for an introduction to proof course that utilizes an IBL approach. The primary objectives of this book are to:

• Expand the mathematical content knowledge of the reader,
• Provide an opportunity for the reader to experience the profound beauty of mathematics,
• Allow the reader to exercise creativity in producing and discovering mathematics,
• Enhance the ability of the reader to be a robust and persistent problem solver.

Ultimately, this is really a book about productive struggle and learning how to learn. Mathematics is simply the vehicle.

Content

The table of contents is listed below. The book includes more content than one can expect to cover in a single semester/quarter. This allows the instructor/reader to pick and choose the sections that suit their needs and desires.

• Preface
• Acknowledgements
• Chapter 1: Introduction
  • 1.1 What is This Course All About?
  • 1.2 An Inquiry-Based Approach
  • 1.3 Structure of the Notes
  • 1.4 Rights of the Learner
  • 1.5 Some Minimal Guidance
• Chapter 2: Mathematics and Logic
  • 2.1 A Taste of Number Theory
  • 2.2 Introduction to Logic
  • 2.3 Techniques for Proving Conditional Statements
  • 2.4 Introduction to Quantification
  • 2.5 More About Quantification
• Chapter 3: Set Theory
  • 3.1 Sets
  • 3.2 Russell’s Paradox
  • 3.3 Power Sets
  • 3.4 Indexing Sets
  • 3.4 Cartesian Products of Sets
• Chapter 4: Induction
  • 4.1 Introduction to Induction
  • 4.2 More on Induction
  • 4.3 Complete Induction and the Well-Ordering Principle
• Chapter 5: The Real Numbers
  • 5.1 Axioms of the Real Numbers
  • 5.2 Standard Topology of the Real Line
• Chapter 6: Three Famous Theorems
  • 6.1 The Fundamental Theorem of Arithmetic
  • 6.2 The Irrationality of $\sqrt{2}$
  • 6.3 The Infinitude of Primes
• Chapter 7: Relations and Partitions
  • 7.1 Relations
  • 7.2 Equivalence Relations
  • 7.3 Partitions
  • 7.4 Modular Arithmetic
• Chapter 8: Functions
  • 8.1 Introduction to Functions
  • 8.2 Injective and Surjective Functions
  • 8.3 Compositions and Inverse Functions
  • 8.4 Images and Preimages of Functions
  • 8.5 Continuous Real Functions
• Chapter 9: Cardinality
  • 9.1 Introduction to Cardinality
  • 9.2 Finite Sets
  • 9.3 Infinite Sets
  • 9.4 Countable Sets
  • 9.5 Uncountable Sets
• Appendix A: Elements of Style for Proofs
• Appendix B: Fancy Mathematical Terms
• Appendix C: Paradoxes
• Appendix D: Definitions in Mathematics

The following sections form the core of the book and are likely the sections that an instructor would focus on in a one-semester introduction to proof course.

• Chapter 2: Mathematics and Logic. All sections.
• Chapter 3: Set Theory. Sections 3.1, 3.3, 3.4, and 3.5.
• Chapter 4: Induction. All sections.
• Chapter 7: Relations and Partitions. Sections 7.1, 7.2, and 7.3.
• Chapter 8: Functions. Sections 8.1, 8.2, 8.3, and 8.4.
• Chapter 9: Cardinality. All sections.

Time permitting, instructors can pick and choose topics from the remaining sections. I typically cover the core sections listed above together with Chapter 6: Three Famous Theorems during a single semester. The Instructor Guide contains examples of a few possible paths through the material, as well as information about which sections and theorems depend on material earlier in the book.

Acknowledgments

Several instructors and students have provided extremely useful feedback, which has improved the book at each iteration. Moreover, due to the open-source nature of the book, I have been able to incorporate content written by others. Below is a partial list of people (alphabetical by last name) who have contributed content, advice, or feedback.

• Chris Drupieski, T. Kyle Petersen, and Bridget Tenner (DePaul University). Modifications that these three made to the book inspired me to streamline some of the exposition, especially in the early chapters.
• Paul Ellis (Rutgers University). Paul has provided lots of useful feedback and several suggestions for improvements. Paul suggested problems for Chapter 4 and provided an initial draft of Section 8.4: Images and Preimages of Functions.
• Jason Grout (Bloomberg, L.P.). I am extremely grateful to Jason for feedback on early versions of this manuscript, as well as for helping me with a variety of technical aspects of writing an open-source textbook.
• Anders Hendrickson (Milliman). Anders is the original author of the content in Appendix A: Elements of Style for Proofs. The current version in Appendix A is a result of modifications made by myself with some suggestions from David Richeson.
• Rebecca Jayne (Hampden-Sydney College). The current version of Section 4.3: Complete Induction is a derivative of content originally contributed by Rebecca.
• Matthew Jones (CSU Dominguez Hills) and Stan Yoshinobu (University of Toronto). A few of the sections were originally adaptations of notes written by Matt and Stan. Early versions of this textbook relied heavily on their work. Moreover, Matt and Stan were two of the key players that contributed to shaping my approach to teaching.
• David Richeson (Dickinson College). David is responsible for much of the content in Appendix B: Fancy Mathematical Terms, Appendix C: Paradoxes, and Appendix D: Definitions in Mathematics. In addition, the current version of Chapter 6: Three Famous Theorems is heavily based on content contributed by David.
• Carol Schumacher (Kenyon College). When I was transitioning to an IBL approach to teaching, Carol was one of my mentors and played a significant role in my development as a teacher. Moreover, this work is undoubtedly influenced my Carol’s excellent book Chapter Zero: Fundamental Notions of Advanced Mathematics, which I used when teaching my very first IBL course.
• Josh Wiscons (CSU Sacramento). The current version of Section 7.4: Modular Arithmetic is a derivative of content contributed by Josh.