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An Introduction to Proof via Inquiry-Based Learning

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 question, 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?

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, subject to change!)

• 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 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

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

Miscellaneous Materials

• 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 (1:25min)
  • 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)
  • Productive Failure | Manu Kapur | TEDxLugano (17:28 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
• Do Hard Things | The Learner Lab (2:41 min)
• 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.
  • Problem 2.4(a)
  • Corollary 2.9
  • Theorem 2.10)

Exams

Below are links to the take-home portions of each exam. 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.

• Exam 1 (Part 2) (PDF) (Due by 11:59PM on Tuesday, October 12)
• Exam 2 (Part 2) (PDF) (Due by beginning of class on Wednesday, November 22)
• Final Exam (Part 2) (PDF) (Due by 1pm Friday, December 10)