## Foundations of Math

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, but the first half of the notes are an adaptation of notes written by Stan Yoshinobu (Cal Poly) and Matthew Jones (California State University, Dominguez Hills). 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.

In addition to working the problems in the notes, I expect you to be *reading* them. 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 in class or posting questions to the course forum on Canvas.

You can find the course notes below. 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.

- Elements of Style of Proofs (PDF)
- Chapter 0: Foundations of Higher Mathematics (PDF)
- Chapter 1: Introduction to Mathematics
- Section 1.1: A Taste of Number Theory (1.1-1.21) (PDF)
- Section 1.2: Logic, Negation, and Contrapositive (1.22-1.45) (PDF)
- Section 1.3: Negating Implications and Proof by Contradiction (1.46-1.59) (PDF)
- Section 1.4: Introduction to Quantification (1.60-1.74) (PDF)
- Section 1.5: More on Quantification (1.75-1.87) (PDF)
- Section 1.6: And Even More on Quantification (1.88-1.103) (PDF)

- Chapter 2: Set Theory and Topology
- Section 2.1: Sets (2.1-2.27) (PDF)
- Section 2.2: Power Sets and Paradoxes (2.28-2.44) (PDF)
- Section 2.3: Indexing Sets (2.45-2.60) (PDF)
- Section 2.4: Basic Topology of $\mathbb{R}$ (2.61-2.103) (PDF)

- Chapter 3: Induction
- Section 3.1: Introduction to Induction (3.1-3.10) (PDF)

- Chapter 4: Two Famous Theorems
- Section 4.1: The Infinitude of Primes (4.1-4.5) (PDF)
*Note:*Theorem 4.6 appears in the next section, but it really should have come before 4.5. - Section 4.2: The Irrationality of $\sqrt{2}$ (4.6-4.17) (PDF)

- Section 4.1: The Infinitude of Primes (4.1-4.5) (PDF)
- Chapter 5: Relations and Functions
- Section 5.1: Relations (5.1-5.28) (PDF)
- Section 5.2: Equivalence Relations (5.29-5.47) (PDF)
- Section 5.3: Partitions (5.48-5.68) (PDF)
- Section 5.4: Introduction to Functions (5.69-5.92) (PDF)
- Section 5.5: Compositions and Inverses of Functions (5.93-5.109) (PDF)
- Section 5.6: Order Relations