2. Teaching
3. MAT 320
4. Materials

An Introduction to Proof via Inquiry-Based Learning

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?

Miscellaneous Materials

• Student Contract
• Deep Practice

Exams

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)