Alphabeta Math

Math Library

Definitions, theorems and fully phase-stratified proofs, mechanically cross-referenced.

How a level is built — step 0 to 10 →Authoring a proof, and repairing one →

Abstract Algebra

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30 pages · 15 with a worked companion

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Binary Operations, Monoids, Groups and SubgroupsExamples & counterexamples →0
Cosets, Index and Lagrange's TheoremExamples & counterexamples →1
Rings, Subrings, Integral Domains and FieldsExamples & counterexamples →1
Normal Subgroups and Quotient GroupsExamples & counterexamples →2
Group Homomorphisms and the Isomorphism TheoremsExamples & counterexamples →3
Cyclic Groups and Direct ProductsExamples & counterexamples →4
Free Groups and PresentationsExamples & counterexamples →4
Group Actions, Orbits, Stabilisers and Cayley's TheoremExamples & counterexamples →4
Ideals, Quotient Rings and the Isomorphism Theorems for RingsExamples & counterexamples →4
Divisibility, Euclidean Domains, Principal Ideal Domains and Unique FactorisationExamples & counterexamples →5
Free Products and AmalgamationExamples & counterexamples →5
Modules, Submodules, Quotient Modules and the Isomorphism TheoremsExamples & counterexamples →5
Symmetric Groups, Cycle Decomposition and the Sign HomomorphismExamples & counterexamples →5
The Fundamental Theorem of Finite Abelian GroupsExamples & counterexamples →5
Polynomial Rings, the Division Algorithm and RootsExamples & counterexamples →6

category-theory

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2 pages · 1 with a worked companion

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Categories, Functors and Natural TransformationsExamples & counterexamples →0

Combinatorics

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24 pages · 12 with a worked companion

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Finite Counting, Factorials and Binomial CoefficientsExamples & counterexamples →0
Inclusion–Exclusion, the Pigeonhole Principle and Double CountingExamples & counterexamples →1
Chains, Antichains, Sperner and DilworthExamples & counterexamples →2
Graphs, Walks and ConnectivityExamples & counterexamples →2
Eulerian and Hamiltonian GraphsExamples & counterexamples →3
Graph ColouringExamples & counterexamples →3
Incidence Algebras and Möbius InversionExamples & counterexamples →3
Induced Subgraphs and Hereditary Graph ClassesExamples & counterexamples →3
Ramsey TheoryExamples & counterexamples →3
Trees, Forests and Spanning TreesExamples & counterexamples →3
Matchings, Covers, Menger and Network FlowsExamples & counterexamples →4
Plane Graphs, Euler's Formula and the Five Colour TheoremExamples & counterexamples →5

Foundations

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14 pages · 6 with a worked companion

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The ZFC Axioms and the Basic Set ConstructionsExamples & counterexamples →0
Relations, Functions, and QuotientsExamples & counterexamples →1
Construction of the Natural Numbers2
Order, Zorn's Lemma, and the Axiom of ChoiceExamples & counterexamples →3
Filters and UltrafiltersExamples & counterexamples →4
Ordinals, Cardinals, and Transfinite Recursion4
Ordinal Arithmetic and the First Uncountable OrdinalExamples & counterexamples →5
Cardinal Arithmetic, Cofinality and the AlephsExamples & counterexamples →6

Linear Algebra

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12 pages · 6 with a worked companion

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Vector Spaces, Linear Subspaces, Span and Direct SumsExamples & counterexamples →0
Linear Independence, Bases and DimensionExamples & counterexamples →1
Linear Transformations, Rank-Nullity and Quotient SpacesExamples & counterexamples →2
Matrices, the Matrix of a Linear Map, and Change of BasisExamples & counterexamples →3
Gaussian Elimination, Elementary Matrices and Reduced Row Echelon FormExamples & counterexamples →4
Determinants of Matrices over a Commutative RingExamples & counterexamples →5

Recorded, Not Proved Here

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3 pages

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Measure and Integration: Recorded, Not Proved Here0
Set Theory Beyond Choice: Recorded, Not Proved Here0
Open Problems and the Research Frontier1

Number Theory

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6 pages · 3 with a worked companion

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Divisibility, Greatest Common Divisors and Bézout's IdentityExamples & counterexamples →0
Primes, Euclid's Lemma and the Fundamental Theorem of ArithmeticExamples & counterexamples →1
Congruences, the Integers Modulo n and the Chinese Remainder TheoremExamples & counterexamples →2

Real Analysis

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73 pages · 33 with a worked companion

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Construction of the Real Numbers via Cauchy Sequences0
Construction of the Real Numbers via Dedekind Cuts0
Foundations of the Real Numbers for Analysis1
Suprema and InfimaExamples & counterexamples →2
Countability and Uncountability3
Roots, Rational Powers, and Classical Inequalities3
Sequences and Limits4
Monotone Sequences, Bolzano-Weierstrass, and Cauchy CompletenessExamples & counterexamples →5
limsup, liminf, and Subsequential LimitsExamples & counterexamples →6
The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete6
Topology of ℝExamples & counterexamples →6
Equivalent Forms of CompletenessExamples & counterexamples →7
Limits of Real FunctionsExamples & counterexamples →7
Series: Convergence and the Nonnegative TestsExamples & counterexamples →7
Absolute and Conditional Convergence; Rearrangement; ProductsExamples & counterexamples →8
Continuity, IVT, EVT, and Uniform ContinuityExamples & counterexamples →8
The Cantor Set, Baire Category, and Measure Zero in ℝExamples & counterexamples →8
Monotone Functions, Discontinuities, and Continuity SetsExamples & counterexamples →9
The Derivative and the Mean Value TheoremsExamples & counterexamples →10
The Riemann Integral: Definition and IntegrabilityExamples & counterexamples →10
Darboux, L'Hôpital, and Taylor's TheoremExamples & counterexamples →11
Properties of the Integral and the Working FTCExamples & counterexamples →11
Bounded Variation and the Riemann–Stieltjes IntegralExamples & counterexamples →12
ConvexityExamples & counterexamples →12
Improper IntegralsExamples & counterexamples →12
Rⁿ as a Normed Space; Vector-Valued FunctionsExamples & counterexamples →12
Sequences and Series of Functions; Uniform ConvergenceExamples & counterexamples →12
Approximation and Compactness in C(K)Examples & counterexamples →13
Power Series and Real-Analytic FunctionsExamples & counterexamples →13
The Riemann Integral in Rᵐ and Jordan ContentExamples & counterexamples →13
The Total Derivative in ℝᵐ → ℝⁿExamples & counterexamples →13
Sine, Cosine, and the Definition of PiExamples & counterexamples →14
The Exponential FunctionExamples & counterexamples →14
The Inverse and Implicit Function TheoremsExamples & counterexamples →14
Fubini and Change of VariablesExamples & counterexamples →15
Fundamental Trigonometric IdentitiesExamples & counterexamples →15
Mixed Partials, Taylor Formulae, and ExtremaExamples & counterexamples →15
The Logarithm and General PowersExamples & counterexamples →15
Further Trigonometric Identities and Inverse FunctionsExamples & counterexamples →16
The Complex Exponential and Euler's FormulaExamples & counterexamples →16

Topology

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44 pages · 22 with a worked companion

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Metric SpacesExamples & counterexamples →0
Completeness, Completion, and Uniform ContinuityExamples & counterexamples →1
Topological Spaces and ContinuityExamples & counterexamples →1
Compactness in Metric SpacesExamples & counterexamples →2
Subspaces, Products, and QuotientsExamples & counterexamples →2
ConnectednessExamples & counterexamples →3
CompactnessExamples & counterexamples →4
Convergence: Nets and FiltersExamples & counterexamples →5
Countability Axioms and Cardinal FunctionsExamples & counterexamples →5
Function Space Topologies and the Exponential LawExamples & counterexamples →5
Separation Axioms: the HierarchyExamples & counterexamples →5
The Topology of Euclidean SpaceExamples & counterexamples →5
Hausdorff via the DiagonalExamples & counterexamples →6
Homotopy and Homotopy EquivalenceExamples & counterexamples →6
Hereditary and Productive Behaviour of the Separation AxiomsExamples & counterexamples →7
The Fundamental GroupExamples & counterexamples →7
Uniform Spaces: the Three DefinitionsExamples & counterexamples →7
Urysohn's Lemma and the Tietze Extension TheoremExamples & counterexamples →7
Partitions of Unity and ParacompactnessExamples & counterexamples →8
The Tychonoff Embedding and the Stone–Čech CompactificationExamples & counterexamples →8
Metrization: Urysohn, Nagata–Smirnov, Bing, SmirnovExamples & counterexamples →9
Uniform Completeness, Completion, and the Samuel CompactificationExamples & counterexamples →9