Faculty evidence
Lesson Keys
Use keys to evaluate reasoning, not merely final statements. Accept equivalent valid proofs.
U01 L01
Statements, Open Sentences and Mathematical Claims
U01 L02
Definitions as Mathematical Contracts
U01 L03
Examples, Nonexamples and Boundary Cases
U01 L04
Necessary, Sufficient and Equivalent Conditions
U01 L05
Theorems, Lemmas, Corollaries and Conjectures
U01 L06
Counterexamples and the Limits of Evidence
U01 L07
Well-Defined Objects and Ambiguous Notation
U02 L01
Propositions and Logical Connectives
U02 L02
Truth Tables and Logical Equivalence
U02 L03
Implication, Converse, Inverse and Contrapositive
U02 L04
Predicates, Domains and Quantifier Scope
U02 L05
Negating Quantified Statements
U02 L06
Nested Quantifiers and Order Dependence
U02 L07
Rules of Inference and Invalid Argument Forms
U03 L01
Set Equality, Inclusion and Extensional Reasoning
U03 L02
Set Operations and De Morgan Laws
U03 L03
Power Sets, Products and Indexed Families
U03 L04
Functions, Images and Preimages
U03 L05
Injective, Surjective and Bijective Maps
U03 L06
Relations, Equivalence Classes and Partitions
U03 L07
Partial Orders, Maximality and Comparability
U04 L01
Anatomy of a Direct Proof
U04 L02
Parity and Divisibility Proofs
U04 L03
Proving Set Inclusion and Equality
U04 L04
Inequalities and Order Arguments
U04 L05
Function Properties by Direct Argument
U04 L06
Algebraic Identities and Structural Closure
U04 L07
Proof Planning, Backward Analysis and Final Presentation
U05 L01
Proof by Contraposition
U05 L02
Proof by Contradiction
U05 L03
Irrationality and Divisibility Contradictions
U05 L04
Infinitude and Existence by Contradiction
U05 L05
Minimal Counterexamples and Infinite Descent
U05 L06
Contradiction versus Contraposition
U05 L07
Debugging Circular and Hidden-Assumption Proofs
U06 L01
Proof by Exhaustive Cases
U06 L02
Parity, Sign and Congruence Case Splits
U06 L03
Constructive Existence Proofs
U06 L04
Nonconstructive Existence Proofs
U06 L05
Uniqueness Proofs
U06 L06
Existence and Uniqueness Together
U06 L07
Classification Theorems and Edge Cases
U07 L01
Ordinary Mathematical Induction
U07 L02
Summation and Product Identities
U07 L03
Inductive Inequalities and Thresholds
U07 L04
Divisibility and Congruence by Induction
U07 L05
Strong Induction
U07 L06
Well-Ordering and Minimal Counterexample Arguments
U07 L07
Structural Induction on Recursively Defined Objects
U08 L01
Recursive Definitions and Recursive Proofs
U08 L02
Structural Recursion on Strings and Trees
U08 L03
Loop Invariants: Initialization
U08 L04
Loop Invariants: Maintenance and Use
U08 L05
Termination and Well-Founded Measures
U08 L06
Recursive Algorithm Correctness
U08 L07
Proof Obligations, Testing and Formal Verification
U09 L01
The Division Algorithm and Quotient–Remainder Proofs
U09 L02
Greatest Common Divisors and Bézout Identity
U09 L03
Prime Divisibility and Euclid’s Lemma
U09 L04
Unique Factorization Arguments
U09 L05
Congruence Proofs and Modular Arithmetic
U09 L06
Linear Diophantine Equations and Modular Inverses
U09 L07
Infinitely Many Primes and Related Constructions
U10 L01
Absolute Value and Inequality Proofs
U10 L02
Bounds, Suprema and Infima
U10 L03
Sequence Convergence from the Definition
U10 L04
Limit Laws for Sequences
U10 L05
Cauchy Sequences and Completeness
U10 L06
Epsilon–Delta Limits
U10 L07
Continuity and Sequential Characterizations
U11 L01
Proofs from Algebraic Axioms
U11 L02
Substructures and Closure Tests
U11 L03
Bijections and Combinatorial Proofs
U11 L04
Pigeonhole and Extremal Arguments
U11 L05
Double Counting and Inclusion–Exclusion
U11 L06
Graph Proofs with Degrees and Paths
U11 L07
Trees, Cycles and Inductive Graph Proofs
U12 L01
Writing Definitions and Theorem Statements
U12 L02
Proof Architecture and Paragraph-Level Logic
U12 L03
Notation, Display Mathematics and LaTeX Practice
U12 L04
Citations, Attribution and Source Verification
U12 L05
Peer Review and Constructive Referee Reports
U12 L06
Revision from Mathematical Feedback
U12 L07
Research Ethics, AI Use and Reproducibility
U13 L01
Selecting a Feasible Theorem-Level Question
U13 L02
Literature Search and Mathematical Source Map
U13 L03
Definitions, Examples and Conjecture Refinement
U13 L04
Lemma Design and Dependency Planning
U13 L05
Proof Construction and Counterexample Search
U13 L06
Peer Review, Gap Repair and Exposition Revision
U13 L07
Research Paper, Reproducible Appendix and Limitations
U13 L08
Oral Defense and Advanced Analysis Bridge