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Fundamentals of Mathematics — Fall 2026

MATH 2055, Section A · MWF 9:40–10:30, Kalkin 001 · first class Wednesday, September 9
Taylor Dupuy · Innovation E439 · office hours TBA

Lectures

Books

• Main text: Velleman — How To Prove It: A Structured Approach, 3rd edition (Cambridge, 2019)
• Companion: Velleman — How To Prove It with Lean (free online; also as a PDF)
• Supplement: Hammack — Book of Proof
• Supplement: Hutchings — Introduction to Mathematical Arguments
• Supplement: LLM — Mathematics for Computer Science

Links

• LLM — Mathematics for Computer Science (YouTube)
• Lean community (installation instructions and the Zulip chat)
• UVM Mathematics PhD Student Help Sessions (they only advertise as helping up to Calculus but these students know everything and generally are very helpful)
• Other fundamentals of mathematics pages (old exams, old homeworks, solutions)
• Taylor Dupuy — UVM — Spring 2024
• Taylor Dupuy — UVM — Spring 2023
• Taylor Dupuy — UVM — Fall 2017
• Jonathan Sands — UVM — Spring 2017
• John Voight — UVM — Fall 2012
• John Voight — UVM — Fall 2011

Quiz Schedule

• Quiz 01 (Friday, September 18 — "I understand truth tables")
• Quiz 02 (Friday, September 25 — "I understand conditionals, converses, and contrapositives" + "I understand free and bound variables")
• Quiz 03 (Friday, October 2 — "I understand quantifiers and how to negate them")
• Quiz 04 (Friday, October 16 — "I understand how to prove two sets are equal by citing the logical laws")
• Quiz 05 (Friday, October 23 — "I understand proof by contradiction and proof by contrapositive")
• Quiz 06 (Friday, October 30 — "I understand proof by cases" + "I understand existence and uniqueness proofs")
• Quiz 07 (Friday, November 6 — "I understand the ε–δ definition of a limit")
• Quiz 08 (Friday, November 13 — "I understand equivalence relations and modular arithmetic")
• Quiz 09 (Friday, November 20 — "I understand injectivity, surjectivity, and inverses")
• Quiz 10 (Friday, December 4 — "I understand images and inverse images" + "I understand proofs by induction")
• Quiz 11 (Friday, December 11 — "I understand recursion and strong induction")

No quiz Friday, September 11. No class Friday, October 9 (Fall Recess) or November 23, 25 and 27 (Thanksgiving Recess). Chapter 8 is not quizzed.

Lean Days

• Wednesday, October 28 — after Chapter 3
• Friday, December 11 — after Chapters 5 and 6; shares the day with Quiz 11 and §8.3

Before October 28: make a free GitHub account (github.com/signup, two minutes, no cost) and bring a laptop that day if you have one. Chrome or Edge, not Safari or Firefox. No laptop is fine — share with a neighbour. Nothing to install.

These are ordinary class meetings. Nothing in Lean is graded or quizzed. Setup, walkthroughs and exercises are here.

Videos

These are my Math 52 lectures from Fall 2017 (full playlist). That course used Lakins and took the material in a different order, so they are grouped by topic rather than by our schedule, and a few cover things we do not.

Unit I — Propositional Logic

• §1.1 — Lecture 01: Propositions and Formulas
• §1.2 — Lecture 02: Propositional Calculus and De Morgan's Laws
• §1.5 — Lecture 04: Implications

Unit II — Predicate Logic

• §1.3, §2.1 — Lecture 03: Quantifiers
• Beyond us, if you are curious — Lecture 14: Signatures, Formulas, Structures, Theories, and Models

Unit III — Sets

• §1.4 — Lecture 18 (Prof. Rombach): The Empty Set and Set Operations
• §2.3 — Lecture 19: Cartesian Products and Power Sets
• §2.3 (families) — Lecture 21 (with J. Sands): Intersections and Unions of Indexed Families of Sets
• §2.3 (families) — Lecture 22: General Properties of Intersections and Unions over Indexed Sets
• §2.3 (families) — Lecture 20: Infinite Intersections and the Least Upper Bound Property

Chapter 3 — Proofs

• §3.1–3.3 — Lecture 07: Methods of Proof
• §3.3 (divisibility) — Lecture 06: Basic Divisibility Proofs (from axioms)
• §3.6 (uniqueness) — Lecture 08: Using Uniqueness and the Fundamental Theorem of Arithmetic
• Background — Lecture 05: Axioms, Lecture 09: What you can't do with Groups; Special Primes

Chapter 4 — Relations

• §4.5 — Lecture 25: Quotient Maps (Real Projective Line, Modular Arithmetic)
• §4.5 — Lecture 26: Well-Definedness

Chapter 5 — Functions

• §5.1, §5.5 — Lecture 23: Functions (onto, one-to-one, bijective; images, preimages)
• §5.2 — Lecture 24 (with C. Vincent): Injectivity, Surjectivity
• §5.3 — Lecture 27: Composition of Functions and Left/Right Inverses
• §5.3 — Lecture 28: Left and Right Inverses and the Axiom of Choice

Chapter 6 — Mathematical Induction

• §6.1 — Lecture 10: Units; Mathematical Induction
• §6.1 — Lecture 11: Dedekind–Peano Axioms and More Induction
• §6.1 (summations) — Lecture 16: Some Famous Summation Formulas
• §6.4 — Lecture 12: Strong Induction, Nim, and the Fundamental Theorem of Arithmetic
• §6.4 — Lecture 13: Strong Induction iff "Weak" Induction
• Not on our syllabus — Lecture 17: The Binomial Theorem (and Orderings), Lecture 15: Dedekind–Peano vs Peano Arithmetic

Chapter 8 — Infinite Sets

• §8.1 — Lecture 29: Different Size Infinities (Cardinality)
• §8.2 — Lecture 30: Uncountability of the Reals
• §8.2 — Lecture 31: The Power Set of the Naturals; the Cardinality of the Continuum
• §8.3 — Lecture 32: Cantor–Schröder–Bernstein (without proof)
• §8.1 — Lecture 33: Dedekind's Definition of Infinite and Finite Sets
• Bonus — Lecture 32.1: The Pigeonhole Principle — Applications, Lecture 32.2: Proof of the Pigeonhole Principle

Suggested Problems

Velleman, 3rd edition, in the order we cover the sections. Not collected, not graded. The PDF has the same list with commentary on each problem.

Unit I — Propositional Logic

• §1.1 Deductive Reasoning and Logical Connectives — 1, 2, 3, 4, 5, 6, 7, 8, 9
• §1.2 Truth Tables — 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18
• §1.5 The Conditional and Biconditional Connectives — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Unit II — Predicate Logic

• §1.3 Variables and Sets (first part only) — 1, 2
• §2.1 Quantifiers — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• §2.2 Equivalences Involving Quantifiers (logic half) — 1(a)(b)(d), 2(a)(b), 3, 4, 5, 6, 7, 9, 15

Unit III — Sets

• §1.3 Variables and Sets (rest) — 3, 4, 5, 6, 7, 8, 9
• §1.4 Operations on Sets — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17
• §2.2 Equivalences Involving Quantifiers (set half) — 1(c), 2(c), 8, 10, 11, 12, 13, 14
• §2.3 More Operations on Sets — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

Chapter 3 — Proofs

• §3.1 Proof Strategies — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17
• §3.2 Proofs Involving Negations and Conditionals — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18
• §3.3 Proofs Involving Quantifiers — 1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22
• §3.4 Proofs Involving Conjunctions and Biconditionals — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 24, 25, 27
• §3.5 Proofs Involving Disjunctions — 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 20, 27, 31
• §3.6 Existence and Uniqueness Proofs — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
• §3.7 More Examples of Proofs — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Chapter 4 — Relations

• §4.1 Ordered Pairs and Cartesian Products — 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15
• §4.2 Relations — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15
• §4.3 More About Relations — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 22, 24
• §4.5 Equivalence Relations — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 20, 22, 24 (skip 25, 26, 27)

Chapter 5 — Functions

• §5.1 Functions — 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 18, 19(a)(c), 20, 21, 22 (skip 19(b))
• §5.2 One-to-One and Onto — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 18, 19, 20 (skip 22(b), 22(g))
• §5.3 Inverses of Functions — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18
• §5.5 Images and Inverse Images: A Research Project — 1(a)(b)(c), 2(a)(b)(c)(d), 3, 4, 6, 7 (skip 5)

Chapter 6 — Mathematical Induction

• §6.1 Proof by Mathematical Induction — 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
• §6.3 Recursion — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20
• §6.4 Strong Induction — 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20(a)

Chapter 8 — Infinite Sets

• §8.1 Equinumerous Sets — 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 18, 20, 22(a), 26, 29 (skip 2(b), 16(b)(c), 19, 24(b))
• §8.2 Countable and Uncountable Sets — 1, 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18 (skip 3)
• §8.3 The Cantor–Schröder–Bernstein Theorem — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14 (skip 11)

Notes

• Nothing yet.