Starting with elements, subsets, and power sets, this lecture culminates in Cantor's diagonal proof that no set is as large as its own power set.
The even-square and odd-sum examples show how arbitrary choices, assumptions, witnesses, and a want-to-show become a checkable direct proof.
This lecture identifies exactly when an implication is false, then turns quantified negation, contraposition, and contradiction into checkable proof tools.
Propositional logic abstracts English statements into Boolean variables, then uses truth tables to check connectives, translation direction, and equivalences.
This lecture extends propositional logic into a language about objects: distinguish constants, predicates, functions, and propositions, then express some and every with existential and universal quantifiers.
Translate natural language one layer at a time: identify universal and existential forms, then handle quantifier order, negation, restricted quantifiers, and uniqueness.
A function is more than a formula: domain, codomain, totality, and determinism are essential, while the quantifiers defining involutions, injections, and surjections dictate their proofs.
This lecture uses surjections and a proof about birds to separate assuming from proving, then shows that involutions are injective and surjective and carries those ideas into function composition.
Two sets have equal cardinality when a bijection pairs their elements; Cantor's diagonal set defeats every function from S to its power set by constructing a value it misses.
This lecture moves from the formal definitions of graphs and digraphs to independent sets, vertex covers, and their complement relationship.
Starting with walks, paths, cycles, and components, this lecture proves that a graph or its complement is connected and develops the pigeonhole principle through degrees and monochromatic triangles.
Use the generalized pigeonhole principle to force a monochromatic triangle at a six-person party, then solve a movie-preference puzzle through average load and contradiction.
Induction is not a list of checked examples: establish a true starting point, prove that an arbitrary true case transmits truth to the next case, and invoke the induction principle.
This lecture connects starting from ordinary induction to induction may start later, following the official examples and proof obligations.
This lecture connects why begin with a weak computer to from device behavior to a state machine, following the official examples and proof obligations.
This lecture connects the dfa definition connects the first half of cs103 to regular means that some dfa exists, following the official examples and proof obligations.
This lecture connects the automata ladder measures power with languages to dfa transition tables, following the official examples and proof obligations.
This lecture connects from closure properties to a language syntax to regex is mathematics, not one library, following the official examples and proof obligations.
This lecture connects four equivalent descriptions of regularity to the precise finite-memory intuition, following the official examples and proof obligations.
This lecture connects from finite-state limits to recursion to the arithmetic grammar, following the official examples and proof obligations.
This lecture connects why the model changes after cfgs to long addition and local access, following the official examples and proof obligations.
This lecture connects the sample tm looks back from the end to beyond pairwise marking, following the official examples and proof obligations.
This lecture connects a quick quantifier audit for recognizers and deciders to why every decision problem can be represented as a language, following the official examples and proof obligations.
This lecture connects returning from r, re, and utm to three self-reference warm-ups, following the official examples and proof obligations.
This lecture connects defining and locating halt to why halt is recognizable, following the official examples and proof obligations.
This lecture connects the lava diagram's two classification tasks to the deck's operational reading of rice's theorem, following the official examples and proof obligations.
This lecture connects decidable does not mean feasible to efficiency requires choosing a resource, following the official examples and proof obligations.
The final deck reconnects proofs, graphs, automata, and computability, then maps those foundations to Stanford courses that use them.