The School of Numbers & Logic · mathematics, pure and applied
Discrete Math & Formal Logic
Sets, proofs, graphs, and formal logic — the exact, finite mathematics beneath every program and every airtight argument.
How mathematicians actually argue — direct proof, contradiction, and induction, practiced until they feel like handwriting.
Syllabus · 5 units · ~24 hours
Unit I — Statements
Propositions, connectives, and truth tables · Implication and its contrapositive · Quantifiers: for all, there exists · Negating a statement correctly
Unit II — Direct Proof
Definitions as the raw material of proof · Proving statements about even, odd, and divisible · Cases, and the discipline of covering them all · Counterexamples: disproving with one honest witness
Unit III — Indirect Proof
Proof by contrapositive · Proof by contradiction · The irrationality of the square root of two · Choosing the right method for the statement
Unit IV — Induction
The principle of mathematical induction · Base case and inductive step, done carefully · Strong induction · Common induction errors and their repair
Unit V — Sets & Functions
Set operations and proofs of set identities · Injections, surjections, bijections · Proving two sets have the same size
Counting, recursion, relations, and Boolean algebra — the standard mathematical toolkit of the working programmer.
Syllabus · 5 units · ~40 hours
Unit I — Counting
The sum and product rules · Permutations and combinations · The pigeonhole principle · Counting passwords, paths, and possibilities
Unit II — Recursion & Recurrences
Recursive definitions and structural thinking · The Fibonacci recurrence and its closed form · Solving linear recurrences · Recursion trees and the cost of divide-and-conquer
Unit III — Relations & Orders
Relations and their properties · Equivalence relations and partitions · Partial orders and Hasse diagrams · Modular arithmetic as an equivalence in daily use
Unit IV — Boolean Algebra
Boolean expressions and truth functions · Simplification and Karnaugh maps · Logic gates: algebra you can solder
Unit V — Graphs, Briefly
Graphs as models of networks and dependencies · Trees and their uses in computing · Breadth-first and depth-first traversal, mathematically
Deduction as a formal system — syntax, semantics, and the celebrated theorems about what proof can and cannot reach.
Syllabus · 4 units · ~30 hours
Unit I — Propositional Logic
Syntax: well-formed formulas · Semantics: truth assignments and tautologies · Logical equivalence and normal forms
Unit II — Natural Deduction
Inference rules and derivations · Conditional and indirect derivation · Soundness: no lies from true premises
Unit III — First-Order Logic
Predicates, quantifiers, and variables · Translating English into first-order sentences · Models and interpretation · Validity and counterexample structures
Unit IV — The View from Above
Completeness, stated and explained · The compactness theorem and a strange consequence · Gödel's incompleteness theorems, told without folklore · What logic gives computer science
Dots and lines with consequences — connectivity, colorings, matchings, and the routes and networks they explain.
Syllabus · 5 units · ~28 hours
Unit I — Graphs & Degrees
Graphs, vertices, edges, and standard families · The handshake lemma · Isomorphism: same graph, different drawing · Representing graphs for computation
Unit II — Walks & Connectivity
Paths, cycles, and connectedness · Eulerian circuits and the bridges of Königsberg · Hamiltonian cycles and why they are harder · Shortest paths and Dijkstra's algorithm
Unit III — Trees
Characterizations of trees · Spanning trees · Minimum spanning trees: Kruskal and Prim · Counting labeled trees: Cayley's formula
Unit IV — Colorings & Matchings
Vertex coloring and scheduling conflicts · The four color theorem and its story · Bipartite graphs and matchings · Hall's marriage theorem
Unit V — Flows
Networks with capacities · The max-flow min-cut theorem · Applications: assignment, transport, and bottlenecks
Counting past the obvious — inclusion-exclusion, generating functions, and the Ramsey guarantee that order always appears.
Syllabus · 4 units · ~26 hours
Unit I — Counting, Revisited
Bijective proofs: counting by matching · Binomial coefficients and their identities · The binomial theorem · Multisets and stars-and-bars
Unit II — Inclusion–Exclusion
Overcounting and correcting for it · Derangements: the hat-check problem · Counting with forbidden positions
Unit III — Generating Functions
Power series as bookkeeping devices · Solving recurrences with generating functions · Partitions of an integer · The Catalan numbers and their many disguises
Unit IV — Existence Arguments
The pigeonhole principle, sharpened · Ramsey numbers: complete disorder is impossible · The probabilistic method, in one worked example
Cantor's discovery that infinities come in sizes — countability, the diagonal argument, and the axioms that keep paradox out.
Syllabus · 3 units · ~14 hours
Unit I — Comparing Sizes
One-to-one correspondence as the meaning of same size · Hilbert's hotel: infinite sets that swallow guests · The integers and rationals are countable
Unit II — The Diagonal
Cantor's diagonal argument · The reals are uncountable · The power set theorem: always a bigger infinity
Unit III — Paradise & Paradox
Russell's paradox and the crisis it caused · The Zermelo–Fraenkel axioms, in plain terms · The axiom of choice and its strange consequences · The continuum hypothesis: a question with no answer