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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 Wednesday 8:30–9:20AM

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" — §1.2 only: equivalence, and tautology versus contradiction. Nothing from §1.1, and nothing to translate from English.)
• Quiz 02 (Wednesday, September 30 — "I understand conditionals, converses, and contrapositives" + "I understand free and bound variables" — moved from September 23.)
• Quiz 03 (Wednesday, September 30 — "I understand quantifiers and how to negate them")
• Quiz 04 (Wednesday, October 14 — "I understand how to prove two sets are equal by citing the logical laws")
• Quiz 05 (Wednesday, October 21 — "I understand proof by contradiction and proof by contrapositive")
• Quiz 06 (Wednesday, October 28 — "I understand existence and uniqueness proofs" — “exactly one” is a conjunction of two claims, and you prove both halves. §3.5 proof by cases is not on this quiz.)
• Quiz 07 (Wednesday, November 4 — "I understand the ε–δ definition of a limit")
• Quiz 08 (Wednesday, November 11 — "I understand equivalence relations and modular arithmetic")
• Quiz 09 (Wednesday, November 18 — "I understand well-definedness on a quotient" + "I understand surjectivity and inverses")
• Quiz 10 (Wednesday, December 2 — "I understand images and inverse images" + "I can state and prove the geometric sum formula" — you will be asked to state the formula as well as prove it, so know it cold. See §6.3 #5.)
• Quiz 11 (Wednesday, December 9 — "I understand recursion and strong induction")

Quiz 1 was on Friday, September 18. Beginning with Quiz 2, quizzes are on Wednesdays. Quiz 2 has been moved from September 23 to September 30, so Quizzes 2 and 3 are both on Wednesday, September 30. No class Friday, October 9 (Fall Recess) or November 23, 25 and 27 (Thanksgiving Recess). Chapter 8 is not quizzed.

Final exam. There is a final exam, in the Registrar's slot: Monday, December 14, 2026, 10:30 AM–1:15 PM. This is the quiz redo session. The syllabus is the authority on this.

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

Suggested Problems and Videos

Velleman, 3rd ed.; ungraded. The PDF adds commentary. Fall 2017 videos use Lakins, so they are matched by topic, not section.

On the “analyze the logical form” exercises. I do not much like them, and I am assigning almost none of them. Translating Neither Alice nor Bob is in the room into symbols tests your ear for English more than your grasp of logic, and people argue about the answers. This applies to §1.1 #1–8, and to §1.3 #1–2, §1.5 #1–3 and §2.1 #1–3, which stay on the list below only because reading a quantifier correctly is a real skill and there is no other drill for it. Do not spend your week on them. No quiz will ask you to translate an English paragraph into symbols.

§1.2 is where Quiz 1 comes from. §1.1 is vocabulary — ∧, ∨, ¬, premise, conclusion. §1.2 is the first real technique in the course and the only one that always works. Four words you need: two formulas are equivalent if their columns agree in every row (to show they are not, point at one row where they differ); a formula is a tautology if its column is T in every row (P ∨ ¬P); a contradiction if its column is F in every row (P ∧ ¬P); and neither otherwise, which is the ordinary case and is not a failure to answer. An argument is valid if the conclusion is true in every row where all the premises are true — every such row, not merely some row where everything happens to come out true.

Unit I — Propositional Logic

• §1.1 Deductive Reasoning and Logical Connectives — 9 only. Do it before §1.2, and guess at the validity of each argument even when you are not sure — §1.2 #7 is where you check those guesses. #1–8 are the translation exercises and are not assigned.
  • Video: Propositions and Formulas
• §1.2 Truth Tables — 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18. If you are short of time, do 1, 8, 9, 10 — they are the whole of Quiz 1 — and then 7. On #10: we check the first De Morgan law in class, so #10(a) is the one we did not do; do it yourself and you will have verified both.
  • Video: Propositional Calculus and De Morgan's Laws
• §1.5 The Conditional and Biconditional Connectives — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  • Video: Implications

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
  • Video for §1.3 and §2.1: Quantifiers
• §2.2 Equivalences Involving Quantifiers (logic half) — 1(a)(b)(d), 2(a)(b), 3, 4, 5, 6, 7, 9, 15
• Optional (beyond the syllabus): Signatures, Formulas, Structures, Theories, and Models

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
  • Video: The Empty Set and Set Operations (Prof. Rombach)
• §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
  • Videos: Cartesian Products and Power Sets; Intersections and Unions of Indexed Families of Sets (with J. Sands); General Properties of Intersections and Unions over Indexed Sets; Infinite Intersections and the Least Upper Bound Property

Chapter 3 — Proofs

• §3.1 Proof Strategies — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17
  • Video for §3.1–3.3: Methods of Proof
• §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
  • Video: Basic Divisibility Proofs (from axioms)
• §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
  • Video: Using Uniqueness and the Fundamental Theorem of Arithmetic
• §3.7 More Examples of Proofs — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
• Background videos: Axioms; What you can't do with Groups; Special Primes

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)
  • Videos: Quotient Maps (Real Projective Line, Modular Arithmetic); Well-Definedness

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))
  • Video for §5.1 and §5.5: Functions (onto, one-to-one, bijective; images, preimages)
• §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))
  • Video: Injectivity, Surjectivity (with C. Vincent)
• §5.3 Inverses of Functions — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18
  • Videos: Composition of Functions and Left/Right Inverses; Left and Right Inverses and the Axiom of Choice
• §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. #7 is the geometric sum with r = 3 and #13 is the difference of powers; with §6.3 #5 they are the three that build toward Quiz 10.
  • Videos: Units; Mathematical Induction; Dedekind–Peano Axioms and More Induction; Some Famous Summation Formulas
• §6.3 Recursion — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 20. #5 is the geometric sum formula — the one formula on Quiz 10 you have to be able to state from memory. The hypothesis r ≠ 1 is not optional.
• §6.4 Strong Induction — 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20(a)
  • Videos: Strong Induction, Nim, and the Fundamental Theorem of Arithmetic; Strong Induction iff "Weak" Induction
• Optional (not on our syllabus): The Binomial Theorem (and Orderings); Dedekind–Peano vs Peano Arithmetic

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))
  • Videos: Different Size Infinities (Cardinality); Dedekind's Definition of Infinite and Finite Sets
• §8.2 Countable and Uncountable Sets — 1, 2, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18 (skip 3)
  • Videos: Uncountability of the Reals; The Power Set of the Naturals; the Cardinality of the Continuum
• §8.3 The Cantor–Schröder–Bernstein Theorem — 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14 (skip 11)
  • Video: Cantor–Schröder–Bernstein (without proof)
• Bonus videos: The Pigeonhole Principle — Applications; Proof of the Pigeonhole Principle

Notes

• Exercises §1.1 #9(a) and §1.2 #7 — Jane, Pete, and the prizes