History
How sets are defined and set notation
Membership
The empty set
Singleton sets
Subsets
Euler and Venn diagrams
Special sets of numbers in mathematics
Functions
Cardinality
Power sets
Partitions
Basic operations
Applications
Principle of inclusion and exclusion


Set (mathematics)
History
Naive set theory


Set (mathematics)
History
Axiomatic set theory


Set (mathematics)
How sets are defined and set notation

Roster notation
Semantic definition
Set-builder notation
Classifying methods of definition


Set (mathematics)
How sets are defined and set notation
Roster notation


Set (mathematics)
How sets are defined and set notation
Roster notation
Infinite sets in roster notation


Set (mathematics)
How sets are defined and set notation
Semantic definition


Set (mathematics)
How sets are defined and set notation
Set-builder notation


Set (mathematics)
How sets are defined and set notation
Classifying methods of definition


Set (mathematics)
Euler and Venn diagrams


Set (mathematics)
Special sets of numbers in mathematics


Infinite sets and infinite cardinality
The continuum hypothesis


Set (mathematics)
Cardinality
Infinite sets and infinite cardinality


Set (mathematics)
Cardinality
The continuum hypothesis


Set (mathematics)
Principle of inclusion and exclusion


Mathematical object
List of mathematical objects by branch


Defining the ordered pair using set theory


Ordered pair
Informal and formal definitions


Defining the ordered pair using set theory

Wiener's definition
Hausdorff's definition
Kuratowski's definition
Quine–Rosser definition
Cantor–Frege definition
Morse definition
Axiomatic definition


Proving that definitions satisfy the characteristic property


Unions, intersections, and relative complements


Paradoxes
Cantor's theory
Axiomatic theories
Consistency
Utility


Naive set theory
Method
Axiomatic theories


Naive set theory
Sets, membership and equality

Note on consistency
Membership
Equality
Empty set


Naive set theory
Sets, membership and equality
Note on consistency


Naive set theory
Sets, membership and equality
Membership


Naive set theory
Sets, membership and equality
Equality


Naive set theory
Sets, membership and equality
Empty set


Naive set theory
Universal sets and absolute complements


Naive set theory
Ordered pairs and Cartesian products


Naive set theory
Paradoxes in early set theory


Terminology
Arithmetic
Continued fraction representation
Other representations
Formal construction
Properties
Real numbers and topological properties
p-adic numbers


Irreducible fraction
Embedding of integers
Equality
Ordering
Addition
Subtraction
Multiplication
Inverse
Division
Exponentiation to integer power


Rational number
Arithmetic
Irreducible fraction


Rational number
Arithmetic
Embedding of integers


Rational number
Arithmetic
Multiplication


Rational number
Arithmetic
Exponentiation to integer power


Rational number
Continued fraction representation


Rational number
Real numbers and topological properties


History
Basic concepts and notation
Ontology
Formalized set theory
Applications
Areas of study
Objections to set theory
Set theory in mathematical education


Combinatorial set theory
Descriptive set theory
Fuzzy set theory
Inner model theory
Large cardinals
Determinacy
Forcing
Cardinal invariants
Set-theoretic topology


Set theory
Areas of study
Combinatorial set theory


Set theory
Areas of study
Descriptive set theory


Set theory
Areas of study
Fuzzy set theory


Set theory
Areas of study
Inner model theory


Set theory
Areas of study
Large cardinals


Set theory
Areas of study
Cardinal invariants


Set theory
Areas of study
Set-theoretic topology


Set theory
Set theory in mathematical education


Properties
In category theory
Definition by indicator functions
Definition in Principia Mathematica


Singleton (mathematics)
In category theory


Singleton (mathematics)
Definition by indicator functions


Singleton (mathematics)
Definition in Principia Mathematica


Statements and proofs
Discussion and consequences
Historical notes
References
External links


Cantor's paradox
Discussion and consequences


Relation between Euler and Venn diagrams


Relation between Euler and Venn diagrams

Example: Euler- to Venn-diagram and Karnaugh map


Example: Euler- to Venn-diagram and Karnaugh map


Blackboard bold
History
Use in textbooks


"Definition" as anticipation of definition


Well-defined expression
Independence of representative


Well-defined expression
Independence of representative
Functions with one argument


Well-defined expression
Independence of representative
Operations


Well-defined expression
Well-defined notation


Well-defined expression
Other uses of the term


Formal notation
History
Informal proof
Formal proof
Generalising De Morgan duality
Extension to predicate and modal logic
In intuitionistic logic
In computer engineering


Substitution form
Set theory and Boolean algebra
Engineering
Text searching


De Morgan's laws
Formal notation
Substitution form


De Morgan's laws
Formal notation
Set theory and Boolean algebra

Unions and intersections of any number of sets


Unions and intersections of any number of sets


De Morgan's laws
Formal notation
Engineering


De Morgan's laws
Formal notation
Text searching


Negation of a disjunction
Negation of a conjunction


De Morgan's laws
Informal proof
Negation of a disjunction


De Morgan's laws
Informal proof
Negation of a conjunction


De Morgan's laws
Formal proof
Conclusion


De Morgan's laws
Generalising De Morgan duality


De Morgan's laws
Extension to predicate and modal logic


De Morgan's laws
In intuitionistic logic


De Morgan's laws
In computer engineering


Examples
Properties
Field
Related fields
Algebraic integers
Special classes
Notes
References


Numbers defined by radicals
Closed-form number


Algebraic number
Related fields
Numbers defined by radicals


Algebraic number
Related fields
Closed-form number


Examples
Paradoxes
Classes in formal set theories
Notes
References
External links


Class (set theory)
Classes in formal set theories


Elements of a mathematical model
Classifications
Construction
Significance in the natural sciences
Some applications
Examples


Mathematical model
Elements of a mathematical model


A priori information
Complexity
Training, tuning, and fitting
Evaluation and assessment


Mathematical model
Construction
A priori information


Mathematical model
Construction
A priori information
Subjective information


Mathematical model
Construction
Complexity


Mathematical model
Construction
Training, tuning, and fitting


Mathematical model
Construction
Evaluation and assessment

Prediction of empirical data
Scope of the model
Philosophical considerations


Mathematical model
Construction
Evaluation and assessment
Prediction of empirical data


Mathematical model
Construction
Evaluation and assessment
Scope of the model


Mathematical model
Construction
Evaluation and assessment
Philosophical considerations


Mathematical model
Significance in the natural sciences


Relation with equivalence, congruence, and isomorphism


Equality (mathematics)
Equality as predicate


Equality (mathematics)
Approximate equality


Equality (mathematics)
Logical definitions


Equality (mathematics)
Equality in set theory

Set equality based on first-order logic with equality

Set equality based on first-order logic without equality


Set equality based on first-order logic with equality


Set equality based on first-order logic without equality


History
Notation
Properties
Generalizations
Formal definitions


Natural number
History
Modern definitions


Relationship between addition and multiplication

Algebraic properties satisfied by the natural numbers


Natural number
Properties
Multiplication


Relationship between addition and multiplication


Algebraic properties satisfied by the natural numbers


Natural number
Formal definitions
Peano axioms


Natural number
Formal definitions
Set-theoretic definition


Theoremhood and truth
Epistemological considerations
Informal account of theorems
Relation with scientific theories
Terminology
Layout
Lore
Theorems in logic


Theorem
Relation with scientific theories


Syntax and semantics
Interpretation of a formal theorem
Theorems and theories


Theorem
Theorems in logic
Syntax and semantics


Theorem
Theorems in logic
Interpretation of a formal theorem


Theorem
Theorems in logic
Theorems and theories


Family
Career
Mathematical work
Philosophical work
Philosophical legacy
Writings


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)

Introduction to Wissenschaftslehre
Metaphysics
Satz an Sich (proposition in itself)
Ideas and objects
Sensation and simple ideas
Logic
Truth
Judgments and cognitions


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Introduction to Wissenschaftslehre


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Metaphysics


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Satz an Sich (proposition in itself)


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Ideas and objects


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Sensation and simple ideas


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Logic


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Truth


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Judgments and cognitions


Bernard Bolzano
Writings
Translations and compilations


Historical context
Foundational crisis
Toward resolution of the crisis


Foundations of mathematics
Historical context

Platonism as a philosophy of mathematics


Foundations of mathematics
Historical context
Ancient Greek mathematics


Platonism as a philosophy of mathematics


Foundations of mathematics
Historical context
Aristotelian realism


Foundations of mathematics
Historical context
Middle Ages and Renaissance


Foundations of mathematics
Historical context
19th century

Real analysis
Group theory
Non-Euclidean geometries
Projective geometry
Boolean algebra and logic
Peano arithmetic


Foundations of mathematics
Historical context
19th century
Real analysis


Foundations of mathematics
Historical context
19th century
Group theory


Foundations of mathematics
Historical context
19th century
Non-Euclidean geometries


Foundations of mathematics
Historical context
19th century
Projective geometry


Foundations of mathematics
Historical context
19th century
Boolean algebra and logic


Foundations of mathematics
Historical context
19th century
Peano arithmetic


Foundations of mathematics
Foundational crisis


Foundations of mathematics
Foundational crisis
Philosophical views

Philosophical consequences of Gödel's completeness theorem


Foundations of mathematics
Foundational crisis
Philosophical views
Formalism


Foundations of mathematics
Foundational crisis
Philosophical views
Intuitionism


Foundations of mathematics
Foundational crisis
Philosophical views
Logicism


Foundations of mathematics
Foundational crisis
Philosophical views
Set-theoretic Platonism


Foundations of mathematics
Foundational crisis
Philosophical views
Indispensability argument for realism


Foundations of mathematics
Foundational crisis
Philosophical views
Rough-and-ready realism


Philosophical consequences of Gödel's completeness theorem


Foundations of mathematics
Foundational crisis
More paradoxes


Foundations of mathematics
Toward resolution of the crisis


History
Comparing sets
Cardinal numbers
Finite, countable and uncountable sets
Infinite sets
Examples and properties
Union and intersection


Definition 1: |A| = |B|
Definition 2: |A| ≤ |B|
Definition 3: |A| < |B|


Cardinality
Comparing sets
Definition 1: |A| = |B|


Cardinality
Comparing sets
Definition 2: |A| ≤ |B|


Cardinality
Comparing sets
Definition 3: |A| < |B|


Cardinality
Finite, countable and uncountable sets


Cardinality
Infinite sets
Cardinality of the continuum


History
Algebraic properties
Order-theoretic properties
Construction
Computer science
Cardinality


Traditional development
Equivalence classes of ordered pairs
Other approaches


Integer
Construction
Traditional development


Integer
Construction
Equivalence classes of ordered pairs


More complex expressions on the left side of the notation


Set-builder notation
Sets defined by enumeration


Set-builder notation
Sets defined by a predicate


Set-builder notation
Sets defined by a predicate
Specifying the domain


Set-builder notation
Sets defined by a predicate
Examples


Set-builder notation
Equivalent predicates yield equal sets


Set-builder notation
Set existence axiom


Set-builder notation
In programming languages


Definition
Examples
Injections can be undone
Injections may be made invertible
Other properties
Proving that functions are injective
Gallery


Injective function
Injections can be undone


Injective function
Injections may be made invertible


Injective function
Proving that functions are injective


Sketch of a proof that e is transcendental


Transcendental number
Numbers proven to be transcendental


Transcendental number
Possible transcendental numbers


Intensional definition
Extensional definition
History


Extensional and intensional definitions
Intensional definition


Extensional and intensional definitions
Extensional definition


Extensional and intensional definitions
History


Definition
Prototypical examples
Affine structure
Metric structure
Isometries
Topology
Axiomatic definitions
Usage
Other geometric spaces


History of the definition
Motivation of the modern definition
Technical definition


Euclidean space
Definition
History of the definition


Euclidean space
Definition
Motivation of the modern definition


Euclidean space
Definition
Technical definition


Subspaces
Lines and segments
Parallelism


Euclidean space
Affine structure
Subspaces


Euclidean space
Affine structure
Lines and segments


Euclidean space
Affine structure
Parallelism


Distance and length
Orthogonality
Angle
Cartesian coordinates
Other coordinates


Euclidean space
Metric structure
Distance and length


Euclidean space
Metric structure
Orthogonality


Euclidean space
Metric structure
Cartesian coordinates


Euclidean space
Metric structure
Other coordinates


Isometry with prototypical examples
Euclidean group


Euclidean space
Isometries
Isometry with prototypical examples


Euclidean space
Isometries
Euclidean group


Affine space
Projective space
Non-Euclidean geometries
Curved spaces
Pseudo-Euclidean space


Euclidean space
Other geometric spaces
Affine space


Euclidean space
Other geometric spaces
Projective space


Euclidean space
Other geometric spaces
Non-Euclidean geometries


Euclidean space
Other geometric spaces
Curved spaces


Euclidean space
Other geometric spaces
Pseudo-Euclidean space


History
Applications
In calculus
Properties
Representations
Computing the digits
In computer culture
References
Further reading
External links


Compound interest
Bernoulli trials
Standard normal distribution
Derangements
Optimal planning problems
Asymptotics


e (mathematical constant)
Applications
Compound interest


e (mathematical constant)
Applications
Bernoulli trials


e (mathematical constant)
Applications
Standard normal distribution


e (mathematical constant)
Applications
Derangements


e (mathematical constant)
Applications
Optimal planning problems


e (mathematical constant)
Applications
Asymptotics


e (mathematical constant)
In calculus
Alternative characterizations


Calculus
Inequalities
Exponential-like functions
Number theory
Complex numbers


e (mathematical constant)
Properties
Calculus


e (mathematical constant)
Properties
Inequalities


e (mathematical constant)
Properties
Exponential-like functions


e (mathematical constant)
Properties
Number theory


e (mathematical constant)
Properties
Complex numbers


e (mathematical constant)
Representations


e (mathematical constant)
Representations
Stochastic representations


e (mathematical constant)
Representations
Known digits


e (mathematical constant)
Computing the digits


e (mathematical constant)
In computer culture


e (mathematical constant)
Further reading


e (mathematical constant)
External links


History
Notation
Properties
Generalizations
Formal definitions


Natural number
History
Modern definitions


Relationship between addition and multiplication

Algebraic properties satisfied by the natural numbers


Natural number
Properties
Multiplication


Relationship between addition and multiplication


Algebraic properties satisfied by the natural numbers


Natural number
Formal definitions
Peano axioms


Natural number
Formal definitions
Set-theoretic definition


Definition
Representation of relations
Properties of relations
Operations on relations
Examples
Generalizations


Relation (mathematics)
Representation of relations


Relation (mathematics)
Properties of relations


Relation (mathematics)
Properties of relations
Combinations of properties


Relation (mathematics)
Operations on relations


Example
Properties
Representing subsets as functions
Relation to binomial theorem
Recursive definition
Subsets of limited cardinality
Power object
Functors and quantifiers


Power set
Representing subsets as functions


Power set
Subsets of limited cardinality


Aleph-nought
Aleph-one
Continuum hypothesis
Aleph-omega
Aleph-α for general α
Fixed points of omega
Role of axiom of choice


Terminology
Arithmetic
Continued fraction representation
Other representations
Formal construction
Properties
Real numbers and topological properties
p-adic numbers


Irreducible fraction
Embedding of integers
Equality
Ordering
Addition
Subtraction
Multiplication
Inverse
Division
Exponentiation to integer power


Rational number
Arithmetic
Irreducible fraction


Rational number
Arithmetic
Embedding of integers


Rational number
Arithmetic
Multiplication


Rational number
Arithmetic
Exponentiation to integer power


Rational number
Continued fraction representation


Rational number
Real numbers and topological properties


In mathematics
In physics
In computer graphics and spatial data
More dimensions
List of topics by dimension


Vector spaces
Manifolds
Varieties
Krull dimension
Topological spaces
Hausdorff dimension
Hilbert spaces


Dimension
In mathematics
Manifolds
Complex dimension


Dimension
In mathematics
Krull dimension


Dimension
In mathematics
Topological spaces


Dimension
In mathematics
Hausdorff dimension


Spatial dimensions
Time
Additional dimensions


Dimension
In physics
Additional dimensions


Dimension
In computer graphics and spatial data


Minimal model of set theory is countable


Characterizations
Properties
Examples
Without the axiom of choice


Uncountable set
Without the axiom of choice


Definition
Examples
Properties
Space of surjections
Gallery


Surjections as right invertible functions

Cardinality of the domain of a surjection

Induced surjection and induced bijection


Surjections as right invertible functions


Surjective function
Properties
Surjections as epimorphisms


Surjective function
Properties
Surjections as binary relations


Cardinality of the domain of a surjection


Surjective function
Properties
Composition and decomposition


Induced surjection and induced bijection


Surjective function
Space of surjections


Arguments for and against the continuum hypothesis


Continuum hypothesis
Cardinality of infinite sets


Continuum hypothesis
Independence from ZFC


Continuum hypothesis
The generalized continuum hypothesis

Implications of GCH for cardinal exponentiation


Implications of GCH for cardinal exponentiation


Set-theoretic definition
Examples
Most common implementation (set theory)
Cartesian products of several sets
Other forms
Definitions outside set theory


Cartesian product
Set-theoretic definition


A deck of cards
A two-dimensional coordinate system


Cartesian product
Examples
A deck of cards


Cartesian product
Examples
A two-dimensional coordinate system


Cartesian product
Most common implementation (set theory)

Non-commutativity and non-associativity
Intersections, unions, and subsets
Cardinality


Cartesian product
Most common implementation (set theory)
Non-commutativity and non-associativity


Cartesian product
Most common implementation (set theory)
Intersections, unions, and subsets


Cartesian product
Most common implementation (set theory)
Cardinality


Cartesian product
Cartesian products of several sets

n-ary Cartesian product
n-ary Cartesian power
Infinite Cartesian products


Cartesian product
Cartesian products of several sets
n-ary Cartesian product


Cartesian product
Cartesian products of several sets
n-ary Cartesian power


Cartesian product
Cartesian products of several sets
Infinite Cartesian products


Abbreviated form
Cartesian product of functions
Cylinder


Cartesian product
Other forms
Abbreviated form


Cartesian product
Other forms
Cartesian product of functions


Cartesian product
Definitions outside set theory


Cartesian product
Definitions outside set theory
Category theory


Cartesian product
Definitions outside set theory
Graph theory


Union of two sets
Algebraic properties
Finite unions
Arbitrary unions
Notation encoding


Union (set theory)
Arbitrary unions
Notations


Absolute complement
Relative complement
Complementary relation
LaTeX notation
In programming languages


Complement (set theory)
Absolute complement


Complement (set theory)
Absolute complement
Definition


Complement (set theory)
Absolute complement
Examples


Complement (set theory)
Absolute complement
Properties


Complement (set theory)
Relative complement


Complement (set theory)
Relative complement
Definition


Complement (set theory)
Relative complement
Examples


Complement (set theory)
Relative complement
Properties


Complement (set theory)
Complementary relation


Complement (set theory)
In programming languages


Definition
Examples
More mathematical examples
Inverses
Composition
Cardinality
Properties
Category theory
Generalization to partial functions
Gallery


Batting line-up of a baseball or cricket team


Bijection
Examples
Seats and students of a classroom


Bijection
Generalization to partial functions


Topological and differential geometric notions


Plane (mathematics)
Euclidean plane
Embedding in three-dimensional space


Plane (mathematics)
Further generalizations


History
Basic concepts and notation
Ontology
Formalized set theory
Applications
Areas of study
Objections to set theory
Set theory in mathematical education


Combinatorial set theory
Descriptive set theory
Fuzzy set theory
Inner model theory
Large cardinals
Determinacy
Forcing
Cardinal invariants
Set-theoretic topology


Set theory
Areas of study
Combinatorial set theory


Set theory
Areas of study
Descriptive set theory


Set theory
Areas of study
Fuzzy set theory


Set theory
Areas of study
Inner model theory


Set theory
Areas of study
Large cardinals


Set theory
Areas of study
Cardinal invariants


Set theory
Areas of study
Set-theoretic topology


Set theory
Set theory in mathematical education


History
Examples
Types
Decimal expansions
Irrational powers
Open questions
In constructive mathematics
Set of all irrationals


Ancient Greece
India
Middle Ages
Modern period


Irrational number
History
Ancient Greece


Irrational number
Examples
General roots


Irrational number
Types
Transcendental/algebraic


Irrational number
In constructive mathematics


Irrational number
Set of all irrationals


Etymology
Historical development
Mathematical logic


Early Greeks
Modern development
Other sciences


Axiom
Historical development
Early Greeks


Axiom
Historical development
Modern development


Axiom
Historical development
Other sciences


Logical axioms
Non-logical axioms
Role in mathematical logic
Further discussion


Axiom
Mathematical logic
Logical axioms
Examples


Axiom
Mathematical logic
Non-logical axioms


Axiom
Mathematical logic
Non-logical axioms
Examples


Axiom
Mathematical logic
Role in mathematical logic


Axiom
Mathematical logic
Role in mathematical logic
Deductive systems and completeness


Axiom
Mathematical logic
Further discussion


Countably infinite sets
Uncountably infinite sets


Infinite set
Examples
Countably infinite sets


Infinite set
Examples
Uncountably infinite sets


Characterizing properties
Arithmetic
Dedekind completeness
Decimal representation
Topological completeness
Cardinality
Other properties
History
Formal definitions
Applications and connections
Vocabulary and notation
Generalizations and extensions


Auxiliary operations
Auxiliary order relations
Integers and fractions as real numbers


Real number
Arithmetic
Auxiliary operations


Real number
Arithmetic
Auxiliary order relations


Real number
Arithmetic
Integers and fractions as real numbers


Real number
Topological completeness
"The complete ordered field"


Axiomatic approach
Construction from the rational numbers


Real number
Formal definitions
Axiomatic approach


Real number
Formal definitions
Construction from the rational numbers


Real number
Applications and connections


Real number
Applications and connections
Physics


Real number
Applications and connections
Logic


Real number
Applications and connections
Computation


Real number
Applications and connections
Set theory


Real number
Generalizations and extensions


Informal presentation
Formal presentation
Philosophical implications
Set-theoretic responses
History
Applied versions
Applications and related topics
Related paradoxes


Russell's paradox
Philosophical implications


Russell's paradox
Set-theoretic responses


Russell's paradox
Applications and related topics


Russell's paradox
Applications and related topics
Russell-like paradoxes


Notation
Properties
In other areas of mathematics
Questioned existence


Empty set
Properties
Operations on the empty set


Extended real numbers
Topology
Category theory
Set theory


Empty set
In other areas of mathematics
Extended real numbers


Empty set
In other areas of mathematics
Topology


Empty set
In other areas of mathematics
Category theory


Empty set
In other areas of mathematics
Set theory


Axiomatic set theory
Philosophical issues


Empty set
Questioned existence
Axiomatic set theory


Empty set
Questioned existence
Philosophical issues


Etymology
Properties
Definitions
n-tuples of m-sets
Type theory


Tuple
Etymology
Names for tuples of specific lengths


Tuples as functions
Tuples as nested ordered pairs
Tuples as nested sets


Tuple
Definitions
Tuples as nested ordered pairs


Notations
Examples
Relation to Boolean algebras
Properties of Boolean rings
Unification


Boolean ring
Relation to Boolean algebras


Boolean ring
Properties of Boolean rings


In the foundations of mathematics and set theory


Total, univalent relation
Set exponentiation


Function (mathematics)
Definition
Total, univalent relation


Function (mathematics)
Definition
Set exponentiation


Functional notation
Arrow notation
Index notation
Dot notation
Specialized notations


Function (mathematics)
Notation
Functional notation


Function (mathematics)
Notation
Arrow notation


Function (mathematics)
Notation
Index notation


Function (mathematics)
Notation
Dot notation


Function (mathematics)
Notation
Specialized notations


Function (mathematics)
Specifying a function

By listing function values
By a formula
Inverse and implicit functions
Using differential calculus
By recurrence


Function (mathematics)
Specifying a function
By listing function values


Function (mathematics)
Specifying a function
By a formula


Function (mathematics)
Specifying a function
Inverse and implicit functions


Function (mathematics)
Specifying a function
Using differential calculus


Function (mathematics)
Specifying a function
By recurrence


Function (mathematics)
Representing a function


Function (mathematics)
Representing a function
Graphs and plots


Function (mathematics)
Representing a function
Tables


Function (mathematics)
Representing a function
Bar chart


Function (mathematics)
General properties

Injective, surjective and bijective functions


Function (mathematics)
General properties
Standard functions


Function (mathematics)
General properties
Function composition


Function (mathematics)
General properties
Image and preimage


Injective, surjective and bijective functions


Function (mathematics)
General properties
Restriction and extension


Function (mathematics)
Multivariate function


Function (mathematics)
In calculus
Real function


Function (mathematics)
In calculus
Vector-valued function


Function (mathematics)
Multi-valued functions


In the foundations of mathematics and set theory


Function (mathematics)
In computer science


Intensional definition
Extensional definition
History


Extensional and intensional definitions
Intensional definition


Extensional and intensional definitions
Extensional definition


Extensional and intensional definitions
History


History
Usage in English
Usage in other languages
Mathematics and logic
Computing


Syntactical-deductive
Syntactical-descriptive
Appositive
Segmental
Use of capitals
Spacing and parentheses


Colon (punctuation)
Usage in English
Syntactical-deductive


Colon (punctuation)
Usage in English
Syntactical-descriptive


Colon (punctuation)
Usage in English
Appositive


Colon (punctuation)
Usage in English
Segmental


Colon (punctuation)
Usage in English
Use of capitals


Colon (punctuation)
Usage in English
Spacing and parentheses


Colon (punctuation)
Usage in other languages

Suffix separator
Abbreviation mark
Word separator
End of sentence or verse
Score divider


Colon (punctuation)
Usage in other languages
Suffix separator


Colon (punctuation)
Usage in other languages
Abbreviation mark


Colon (punctuation)
Usage in other languages
Word separator


Colon (punctuation)
Usage in other languages
End of sentence or verse


Colon (punctuation)
Usage in other languages
Score divider


Colon (punctuation)
Mathematics and logic


Programming languages
Addresses
Text markup


Colon (punctuation)
Computing
Programming languages


Colon (punctuation)
Computing
Text markup


Generalizations
Intersections
Disjoint unions and partitions


Disjoint sets
Disjoint unions and partitions


Usage note
Independence results in set theory
Applications to physical theory


Independence (mathematical logic)
Usage note


Independence (mathematical logic)
Independence results in set theory


Independence (mathematical logic)
Applications to physical theory


Properties
Definition
Other representations
Generalizations of the Euclidean line
Extensions


In higher dimensions
Types
In axiomatic systems


Line (geometry)
Properties
In higher dimensions


Line (geometry)
Properties
In higher dimensions
Collinear points


Line (geometry)
Properties
In axiomatic systems


Linear equation
Parametric equation
Hesse normal form


Line (geometry)
Definition
Linear equation


Line (geometry)
Definition
Parametric equation


Line (geometry)
Definition
Hesse normal form


Line (geometry)
Other representations
Vectors


Line (geometry)
Other representations
Polar coordinates


Line (geometry)
Generalizations of the Euclidean line


Line (geometry)
Generalizations of the Euclidean line
Projective geometry


Statements consistent with the negation of AC


Nomenclature ZF, AC, and ZFC
Variants
Restriction to finite sets


Axiom of choice
Statement
Nomenclature ZF, AC, and ZFC


Axiom of choice
Statement
Restriction to finite sets


Axiom of choice
Criticism and acceptance


Axiom of choice
In constructive mathematics


Axiom of choice
Equivalents
Category theory


Results requiring AC (or weaker forms) but weaker than it


Axiom of choice
Weaker forms
Possibly equivalent implications of AC


Axiom of choice
Stronger forms of the negation of AC


Axiom of choice
Axiom of choice in type theory


Notation and terminology
Definition
Algebraic properties
Arbitrary intersections
Nullary intersection


Intersection (set theory)
Notation and terminology


Intersection (set theory)
Definition
Intersecting and disjoint sets


Intersection (set theory)
Algebraic properties


Intersection (set theory)
Arbitrary intersections


Intersection (set theory)
Nullary intersection


Absolute complement
Relative complement
Complementary relation
LaTeX notation
In programming languages


Complement (set theory)
Absolute complement


Complement (set theory)
Absolute complement
Definition


Complement (set theory)
Absolute complement
Examples


Complement (set theory)
Absolute complement
Properties


Complement (set theory)
Relative complement


Complement (set theory)
Relative complement
Definition


Complement (set theory)
Relative complement
Examples


Complement (set theory)
Relative complement
Properties


Complement (set theory)
Complementary relation


Complement (set theory)
In programming languages


The fundamental properties of set algebra

Some additional laws for unions and intersections


The fundamental properties of set algebra


Algebra of sets
The principle of duality


Some additional laws for unions and intersections


Algebra of sets
Some additional laws for complements


Algebra of sets
The algebra of inclusion


Algebra of sets
The algebra of relative complements


Early life and education
Career
Death
Selected publications


In a specific context
In ordinary mathematics
In set theory
In predicate calculus
In category theory
In type theory


Universe (mathematics)
In a specific context


Universe (mathematics)
In ordinary mathematics


Universe (mathematics)
In predicate calculus


Universe (mathematics)
In category theory


In real or complex vector spaces
Properties
In proofs
As a degenerate ellipse
In other geometric shapes
Directed line segment
Generalizations
Types of line segments


Line segment
In real or complex vector spaces


Triangles
Quadrilaterals
Circles and ellipses


Line segment
In other geometric shapes
Triangles


Line segment
In other geometric shapes
Quadrilaterals


Line segment
In other geometric shapes
Circles and ellipses


Sets
Notation and terminology
Cardinality of sets
Examples
Formal relation


Element (mathematics)
Notation and terminology


Element (mathematics)
Cardinality of sets


Informal presentation
Formal presentation
Philosophical implications
Set-theoretic responses
History
Applied versions
Applications and related topics
Related paradoxes


Russell's paradox
Philosophical implications


Russell's paradox
Set-theoretic responses


Russell's paradox
Applications and related topics


Russell's paradox
Applications and related topics
Russell-like paradoxes


History
Axioms
Motivation via the cumulative hierarchy
Metamathematics
Criticisms


2. Axiom of regularity (also called the axiom of foundation)

3. Axiom schema of specification (or of separation, or of restricted comprehension)


Zermelo–Fraenkel set theory
Axioms
1. Axiom of extensionality


2. Axiom of regularity (also called the axiom of foundation)


3. Axiom schema of specification (or of separation, or of restricted comprehension)


Zermelo–Fraenkel set theory
Axioms
4. Axiom of pairing


Zermelo–Fraenkel set theory
Axioms
5. Axiom of union


Zermelo–Fraenkel set theory
Axioms
6. Axiom schema of replacement


Zermelo–Fraenkel set theory
Axioms
7. Axiom of infinity


Zermelo–Fraenkel set theory
Axioms
8. Axiom of power set


Zermelo–Fraenkel set theory
Axioms
9. Well-ordering axiom


Zermelo–Fraenkel set theory
Axioms
Zorn's lemma


Zermelo–Fraenkel set theory
Motivation via the cumulative hierarchy


Zermelo–Fraenkel set theory
Metamathematics

Virtual classes
Von Neumann–Bernays–Gödel set theory
Consistency
Independence
Proposed additions


Zermelo–Fraenkel set theory
Metamathematics
Virtual classes


Zermelo–Fraenkel set theory
Metamathematics
Von Neumann–Bernays–Gödel set theory


Zermelo–Fraenkel set theory
Metamathematics
Consistency


Zermelo–Fraenkel set theory
Metamathematics
Independence


Zermelo–Fraenkel set theory
Metamathematics
Proposed additions


On Internet chat rooms and in text messaging


In English
In Polish
In Russian
In Japanese
In Chinese
In Spanish
In French
In German
In Italian


Ellipsis
In different languages
In English


Ellipsis
In different languages
In English
American English


Ellipsis
In different languages
In English
British English


Ellipsis
In different languages
In Polish


Ellipsis
In different languages
In Russian


Ellipsis
In different languages
In Japanese


Ellipsis
In different languages
In Chinese


Ellipsis
In different languages
In Spanish


Ellipsis
In different languages
In French


Ellipsis
In different languages
In German


Ellipsis
In different languages
In Italian


Ellipsis
Computer science
Programming languages


History
Typography
Round brackets or parentheses
Square brackets or brackets
Curly brackets or braces
Angle brackets or chevrons
Other brackets
Unicode


Bracket
Round brackets or parentheses
Uses of ( )

Enumerations
Accounting
Parentheses in mathematics
Parentheses in programming languages
Taxonomy
Chemistry and physics
Pop songs


Bracket
Round brackets or parentheses
Uses of ( )
Enumerations


Bracket
Round brackets or parentheses
Uses of ( )
Accounting


Bracket
Round brackets or parentheses
Uses of ( )
Parentheses in mathematics


Bracket
Round brackets or parentheses
Uses of ( )
Parentheses in programming languages


Bracket
Round brackets or parentheses
Uses of ( )
Taxonomy


Bracket
Round brackets or parentheses
Uses of ( )
Chemistry and physics


Bracket
Round brackets or parentheses
Uses of ( )
Pop songs


Bracket
Square brackets or brackets
Uses of [ ]

Square brackets in programming languages


Bracket
Square brackets or brackets
Uses of [ ]
Proofreading


Bracket
Square brackets or brackets
Uses of [ ]
Law


Bracket
Square brackets or brackets
Uses of [ ]
Square brackets in mathematics


Bracket
Square brackets or brackets
Uses of [ ]
Chemistry


Square brackets in programming languages


Bracket
Curly brackets or braces
Uses of { }

Music
Curly brackets in programming languages
Curly brackets in mathematics


Bracket
Curly brackets or braces
Uses of { }
Music


Bracket
Curly brackets or braces
Uses of { }
Curly brackets in programming languages


Bracket
Curly brackets or braces
Uses of { }
Curly brackets in mathematics


Bracket
Angle brackets or chevrons
Shape


Bracket
Angle brackets or chevrons
Uses of ⟨ ⟩

Angle brackets in mathematics
Physics and mechanics
Angle brackets in programming languages


Bracket
Angle brackets or chevrons
Uses of ⟨ ⟩
Angle brackets in mathematics


Bracket
Angle brackets or chevrons
Uses of ⟨ ⟩
Physics and mechanics


Bracket
Angle brackets or chevrons
Uses of ⟨ ⟩
Angle brackets in programming languages


Floor ⌊ ⌋ and ceiling ⌈ ⌉ corner brackets

Quine corners ⌜⌝ and half brackets ⸤ ⸥ or ⸢ ⸣


Bracket
Other brackets
Lenticular brackets【】


Floor ⌊ ⌋ and ceiling ⌈ ⌉ corner brackets


Quine corners ⌜⌝ and half brackets ⸤ ⸥ or ⸢ ⸣


Bracket
Other brackets
Double brackets ⟦ ⟧


Bracket
Other brackets
Brackets with quills ⁅ ⁆


History
Algebraic properties
Order-theoretic properties
Construction
Computer science
Cardinality


Traditional development
Equivalence classes of ordered pairs
Other approaches


Integer
Construction
Traditional development


Integer
Construction
Equivalence classes of ordered pairs


History
Permutations without repetitions
Definition
Notations
Other uses of the term permutation
Properties
Permutations of totally ordered sets
Permutations in computing


Permutation
Permutations without repetitions


Two-line notation
One-line notation
Cycle notation
Canonical cycle notation
Composition of permutations


Permutation
Notations
Canonical cycle notation


Permutation
Notations
Composition of permutations


Permutation
Other uses of the term permutation

k-permutations of n
Permutations with repetition
Permutations of multisets
Circular permutations


Permutation
Other uses of the term permutation
k-permutations of n


Permutation
Other uses of the term permutation
Permutations with repetition


Permutation
Other uses of the term permutation
Permutations of multisets


Permutation
Other uses of the term permutation
Circular permutations


Cycle type
Conjugating permutations
Permutation order
Parity of a permutation
Matrix representation


Permutation
Properties
Conjugating permutations


Permutation
Properties
Permutation order


Permutation
Properties
Parity of a permutation


Permutation
Properties
Matrix representation


Permutation
Permutations of totally ordered sets

Ascents, descents, runs and exceedances
Foata's transition lemma
Inversions


Permutation
Permutations of totally ordered sets
Ascents, descents, runs and exceedances


Permutation
Permutations of totally ordered sets
Foata's transition lemma


Permutation
Permutations of totally ordered sets
Inversions


Numbering permutations
Algorithms to generate permutations
Applications


Permutation
Permutations in computing
Numbering permutations


Permutation
Permutations in computing
Algorithms to generate permutations

Random generation of permutations
Generation in lexicographic order
Generation with minimal changes
Meandric permutations


Permutation
Permutations in computing
Algorithms to generate permutations
Random generation of permutations


Permutation
Permutations in computing
Algorithms to generate permutations
Generation in lexicographic order


Permutation
Permutations in computing
Algorithms to generate permutations
Generation with minimal changes


Permutation
Permutations in computing
Algorithms to generate permutations
Meandric permutations


Permutation
Permutations in computing
Applications


Formal systems: completeness, consistency, and effective axiomatization


Formal systems: completeness, consistency, and effective axiomatization

Effective axiomatization
Completeness
Consistency
Systems which contain arithmetic
Conflicting goals


Gödel's incompleteness theorems
First incompleteness theorem

Syntactic form of the Gödel sentence
Truth of the Gödel sentence
Relationship with the liar paradox
Extensions of Gödel's original result


Gödel's incompleteness theorems
First incompleteness theorem
Syntactic form of the Gödel sentence


Gödel's incompleteness theorems
First incompleteness theorem
Truth of the Gödel sentence


Gödel's incompleteness theorems
First incompleteness theorem
Relationship with the liar paradox


Gödel's incompleteness theorems
First incompleteness theorem
Extensions of Gödel's original result


Gödel's incompleteness theorems
Second incompleteness theorem

Expressing consistency
The Hilbert–Bernays conditions
Implications for consistency proofs


Gödel's incompleteness theorems
Second incompleteness theorem
Expressing consistency


Gödel's incompleteness theorems
Second incompleteness theorem
The Hilbert–Bernays conditions


Gödel's incompleteness theorems
Second incompleteness theorem
Implications for consistency proofs


Gödel's incompleteness theorems
Examples of undecidable statements

Undecidable statements provable in larger systems


Undecidable statements provable in larger systems


Gödel's incompleteness theorems
Relationship with computability


Gödel's incompleteness theorems
Proof sketch for the first theorem

Construction of a statement about "provability"


Gödel's incompleteness theorems
Proof sketch for the first theorem
Arithmetization of syntax


Construction of a statement about "provability"


Gödel's incompleteness theorems
Proof sketch for the first theorem
Diagonalization


Gödel's incompleteness theorems
Proof sketch for the first theorem
Proof via Berry's paradox


Gödel's incompleteness theorems
Proof sketch for the first theorem
Computer verified proofs


Gödel's incompleteness theorems
Proof sketch for the second theorem


Gödel's incompleteness theorems
Discussion and implications

Consequences for logicism and Hilbert's second problem

Appeals to the incompleteness theorems in other fields


Consequences for logicism and Hilbert's second problem


Gödel's incompleteness theorems
Discussion and implications
Minds and machines


Gödel's incompleteness theorems
Discussion and implications
Paraconsistent logic


Appeals to the incompleteness theorems in other fields


Announcement
Generalization and acceptance
Criticisms


Gödel's incompleteness theorems
History
Announcement


Gödel's incompleteness theorems
History
Generalization and acceptance


Gödel's incompleteness theorems
History
Criticisms


Gödel's incompleteness theorems
History
Criticisms
Finsler


Gödel's incompleteness theorems
History
Criticisms
Zermelo


Gödel's incompleteness theorems
History
Criticisms
Wittgenstein


History
Notation
Properties
Generalizations
Formal definitions


Natural number
History
Modern definitions


Relationship between addition and multiplication

Algebraic properties satisfied by the natural numbers


Natural number
Properties
Multiplication


Relationship between addition and multiplication


Algebraic properties satisfied by the natural numbers


Natural number
Formal definitions
Peano axioms


Natural number
Formal definitions
Set-theoretic definition


Design
Symbolism
History
Regimental flags
Naval flags
Colonial flags
Other
Gallery


Flag of France
History
Kingdom of France


Examples and notation
Formal definition and basic properties
Limits and convergence
Series
Use in other fields of mathematics


Examples
Indexing
Defining a sequence by recursion


Sequence
Examples and notation
Defining a sequence by recursion


Sequence
Formal definition and basic properties

Definition
Finite and infinite
Increasing and decreasing
Bounded
Subsequences
Other types of sequences


Sequence
Formal definition and basic properties
Definition


Sequence
Formal definition and basic properties
Finite and infinite


Sequence
Formal definition and basic properties
Increasing and decreasing


Sequence
Formal definition and basic properties
Bounded


Sequence
Formal definition and basic properties
Subsequences


Sequence
Formal definition and basic properties
Other types of sequences


Formal definition of convergence
Applications and important results
Cauchy sequences
Infinite limits


Sequence
Limits and convergence
Formal definition of convergence


Sequence
Limits and convergence
Applications and important results


Sequence
Limits and convergence
Cauchy sequences


Sequence
Limits and convergence
Infinite limits


Sequence
Use in other fields of mathematics

Topology
Analysis
Linear algebra
Abstract algebra
Set theory
Computing
Streams


Sequence
Use in other fields of mathematics
Topology


Sequence
Use in other fields of mathematics
Topology
Product topology


Sequence
Use in other fields of mathematics
Analysis


Sequence
Use in other fields of mathematics
Analysis
Sequence spaces


Sequence
Use in other fields of mathematics
Linear algebra


Sequence
Use in other fields of mathematics
Abstract algebra

Free monoid
Exact sequences
Spectral sequences


Sequence
Use in other fields of mathematics
Abstract algebra
Free monoid


Sequence
Use in other fields of mathematics
Abstract algebra
Exact sequences


Sequence
Use in other fields of mathematics
Abstract algebra
Spectral sequences


Sequence
Use in other fields of mathematics
Set theory


Sequence
Use in other fields of mathematics
Computing


Sequence
Use in other fields of mathematics
Streams


History
Examples
Usage
Oblique type compared to italics
More complex usage


Italic type
Oblique type compared to italics


Italics within italics
Left-leaning italics
Upright italics
Parentheses
Substitutes
OpenType
Web pages
Unicode


Italic type
More complex usage
Italics within italics


Italic type
More complex usage
Left-leaning italics


Italic type
More complex usage
Left-leaning italics
Iranic font style


Italic type
More complex usage
Upright italics


Italic type
More complex usage
Parentheses


Italic type
More complex usage
Substitutes


Italic type
More complex usage
Web pages


Definition
Examples
More mathematical examples
Inverses
Composition
Cardinality
Properties
Category theory
Generalization to partial functions
Gallery


Batting line-up of a baseball or cricket team


Bijection
Examples
Seats and students of a classroom


Bijection
Generalization to partial functions


Definition
Notation
Visualization
History
Relations and operations
Properties
Formal construction
Complex analysis
Applications
Generalizations and related notions


Cartesian complex plane
Polar complex plane
Complex graphs


Complex number
Visualization
Cartesian complex plane


Complex number
Visualization
Polar complex plane


Complex number
Visualization
Polar complex plane
Modulus and argument


Complex number
Visualization
Complex graphs


Multiplication and division in polar form


Complex number
Relations and operations
Equality


Complex number
Relations and operations
Ordering


Complex number
Relations and operations
Conjugate


Complex number
Relations and operations
Addition and subtraction


Complex number
Relations and operations
Multiplication and square


Complex number
Relations and operations
Reciprocal and division


Complex number
Relations and operations
Square root


Complex number
Relations and operations
Exponential function


Complex number
Relations and operations
Exponential function
Functional equation


Complex number
Relations and operations
Exponential function
Euler's formula


Complex number
Relations and operations
Complex logarithm


Complex number
Relations and operations
Exponentiation


Complex number
Relations and operations
Exponentiation
Integer and fractional exponents


Field structure
Solutions of polynomial equations
Algebraic characterization
Characterization as a topological field


Complex number
Properties
Field structure


Complex number
Properties
Solutions of polynomial equations


Complex number
Properties
Algebraic characterization


Complex number
Properties
Characterization as a topological field


Matrix representation of complex numbers


Complex number
Formal construction
Construction as ordered pairs


Complex number
Formal construction
Construction as a quotient field


Complex exponential and related functions


Complex number
Complex analysis
Holomorphic functions


Geometry
Algebraic number theory
Analytic number theory
Improper integrals
Dynamic equations
Linear algebra
In applied mathematics
In physics


Complex number
Applications
Geometry
Shapes


Complex number
Applications
Geometry
Fractal geometry


Complex number
Applications
Geometry
Triangles


Complex number
Applications
Algebraic number theory


Complex number
Applications
Analytic number theory


Complex number
Applications
Improper integrals


Complex number
Applications
Dynamic equations


Complex number
Applications
Linear algebra


Complex number
Applications
In applied mathematics


Complex number
Applications
In applied mathematics
Control theory


Complex number
Applications
In applied mathematics
Signal analysis


Electromagnetism and electrical engineering


Complex number
Applications
In physics
Fluid dynamics


Complex number
Applications
In physics
Quantum mechanics


Complex number
Applications
In physics
Relativity


Complex number
Generalizations and related notions


Characterizations
Properties
Examples
Without the axiom of choice


Uncountable set
Without the axiom of choice


Hausdorff distance vs. symmetric difference


Symmetric difference
n-ary symmetric difference


Symmetric difference
Symmetric difference on measure spaces


Details
Example
History
Overview
Extensions to higher numbers of sets
Related concepts


Venn diagram
Extensions to higher numbers of sets


Venn diagram
Extensions to higher numbers of sets
Edwards–Venn diagrams


Venn diagram
Extensions to higher numbers of sets
Other diagrams


Definition and notation
Examples
Partitions and equivalence relations
Refinement of partitions
Noncrossing partitions
Counting partitions


Partition of a set
Definition and notation


Partition of a set
Partitions and equivalence relations


Partition of a set
Refinement of partitions


Partition of a set
Noncrossing partitions


Definitions
Historical overview
Branches of philosophy
Methods of philosophy
Applied and professional philosophy


Western philosophy
Middle Eastern philosophy
Eastern philosophy
African philosophy
Indigenous American philosophy


Philosophy
Historical overview
Western philosophy


Philosophy
Historical overview
Western philosophy
Ancient era


Philosophy
Historical overview
Western philosophy
Medieval era


Philosophy
Historical overview
Western philosophy
Modern era


Philosophy
Historical overview
Middle Eastern philosophy

Pre-Islamic philosophy
Islamic philosophy


Philosophy
Historical overview
Middle Eastern philosophy
Pre-Islamic philosophy


Philosophy
Historical overview
Middle Eastern philosophy
Islamic philosophy


Philosophy
Historical overview
Eastern philosophy

Indian philosophy
Buddhist philosophy
East Asian philosophy


Philosophy
Historical overview
Eastern philosophy
Indian philosophy


Philosophy
Historical overview
Eastern philosophy
Buddhist philosophy


Philosophy
Historical overview
Eastern philosophy
East Asian philosophy


Philosophy
Historical overview
African philosophy


Philosophy
Historical overview
Indigenous American philosophy


Aesthetics
Ethics
Epistemology
Metaphysics
Logic
Mind and language
Philosophy of science
Political philosophy
Philosophy of religion
Metaphilosophy
Other subdivisions


Philosophy
Branches of philosophy
Aesthetics


Philosophy
Branches of philosophy
Ethics


Philosophy
Branches of philosophy
Epistemology


Philosophy
Branches of philosophy
Metaphysics


Philosophy
Branches of philosophy
Mind and language


Philosophy
Branches of philosophy
Philosophy of science


Philosophy
Branches of philosophy
Political philosophy


Philosophy
Branches of philosophy
Philosophy of religion


Philosophy
Branches of philosophy
Metaphilosophy


Philosophy
Branches of philosophy
Other subdivisions


Philosophy
Methods of philosophy
Philosophical progress


Philosophy
Applied and professional philosophy


Philosophy
Applied and professional philosophy
Women in philosophy


Characterizing properties
Arithmetic
Dedekind completeness
Decimal representation
Topological completeness
Cardinality
Other properties
History
Formal definitions
Applications and connections
Vocabulary and notation
Generalizations and extensions


Auxiliary operations
Auxiliary order relations
Integers and fractions as real numbers


Real number
Arithmetic
Auxiliary operations


Real number
Arithmetic
Auxiliary order relations


Real number
Arithmetic
Integers and fractions as real numbers


Real number
Topological completeness
"The complete ordered field"


Axiomatic approach
Construction from the rational numbers


Real number
Formal definitions
Axiomatic approach


Real number
Formal definitions
Construction from the rational numbers


Real number
Applications and connections


Real number
Applications and connections
Physics


Real number
Applications and connections
Logic


Real number
Applications and connections
Computation


Real number
Applications and connections
Set theory


Real number
Generalizations and extensions


Terminology
Typographical considerations
Bicameral script
Stylistic or specialised usage
Case folding and case conversion
History


Letter case
Typographical considerations


Capitalisation
Exceptional letters and digraphs
Related features


Letter case
Bicameral script
Capitalisation


Letter case
Bicameral script
Exceptional letters and digraphs


Letter case
Bicameral script
Related features


Letter case
Stylistic or specialised usage

Unit symbols and prefixes in the metric system


Letter case
Stylistic or specialised usage
Headings and publication titles


Letter case
Stylistic or specialised usage
Multi-word proper nouns


Unit symbols and prefixes in the metric system


Letter case
Stylistic or specialised usage
Use within programming languages


Letter case
Stylistic or specialised usage
Use within programming languages
Camel case


Letter case
Stylistic or specialised usage
Use within programming languages
Snake case


Letter case
Stylistic or specialised usage
Use within programming languages
Kebab case


Letter case
Stylistic or specialised usage
Studly caps


Letter case
Case folding and case conversion

Unicode case folding and script identification


Unicode case folding and script identification


Letter case
Case folding and case conversion
Methods in word processing


Letter case
Case folding and case conversion
Methods in programming


Vocabulary
Forms of fractions
Arithmetic with fractions
Fractions in abstract mathematics
Algebraic fractions
Radical expressions
Typographical variations
History
In formal education


Reciprocals and the "invisible denominator"


Fraction
Forms of fractions
Simple, common, or vulgar fractions


Fraction
Forms of fractions
Proper and improper fractions


Fraction
Forms of fractions
Decimal fractions and percentages


Fraction
Forms of fractions
Mixed numbers


Fraction
Forms of fractions
Historical notions

Egyptian fraction
Complex and compound fractions


Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


Converting between decimals and fractions


Fraction
Arithmetic with fractions
Equivalent fractions


Fraction
Arithmetic with fractions
Equivalent fractions
Simplifying (reducing) fractions


Fraction
Arithmetic with fractions
Comparing fractions


Fraction
Arithmetic with fractions
Addition


Fraction
Arithmetic with fractions
Addition
Adding unlike quantities


Fraction
Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication

Multiplying a fraction by another fraction

Multiplying a fraction by a whole number


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
Division


Converting between decimals and fractions

Converting repeating decimals to fractions


Converting repeating decimals to fractions


Fraction
Fractions in abstract mathematics


Pedagogical tools
Documents for teachers


Fraction
In formal education
Pedagogical tools


Fraction
In formal education
Documents for teachers


Philosophy, religion, literature and Cantor's mathematics


Youth and studies
Teacher and researcher
Later years and death


Georg Cantor
Biography
Youth and studies


Georg Cantor
Biography
Teacher and researcher


Georg Cantor
Biography
Later years and death


Number theory, trigonometric series and ordinals


Georg Cantor
Mathematical work
Set theory

Absolute infinite, well-ordering theorem, and paradoxes


Georg Cantor
Mathematical work
Set theory
One-to-one correspondence


Georg Cantor
Mathematical work
Set theory
Continuum hypothesis


Absolute infinite, well-ordering theorem, and paradoxes


Restrictions, extensions, and variations

Automated theorem proving and formal methods


Alphabet
Formation rules
Free and bound variables
Example: ordered abelian groups


First-order logic
Syntax
Alphabet
Logical symbols


First-order logic
Syntax
Alphabet
Non-logical symbols


First-order logic
Syntax
Formation rules


First-order logic
Syntax
Formation rules
Terms


First-order logic
Syntax
Formation rules
Formulas


First-order logic
Syntax
Formation rules
Notational conventions


First-order logic
Syntax
Free and bound variables


First-order logic
Syntax
Example: ordered abelian groups


Validity, satisfiability, and logical consequence

First-order theories, models, and elementary classes


First-order logic
Semantics
First-order structures


First-order logic
Semantics
Evaluation of truth values


Validity, satisfiability, and logical consequence


First-order logic
Semantics
Algebraizations


First-order theories, models, and elementary classes


First-order logic
Semantics
Empty domains


Hilbert-style systems and natural deduction


First-order logic
Deductive systems
Rules of inference


First-order logic
Deductive systems
Sequent calculus


First-order logic
Deductive systems
Tableaux method


First-order logic
Deductive systems
Resolution


First-order logic
Deductive systems
Provable identities


First-order logic
Equality and its axioms

First-order logic without equality
Defining equality within a theory


First-order logic
Equality and its axioms
First-order logic without equality


First-order logic
Equality and its axioms
Defining equality within a theory


First-order logic
Metalogical properties

Completeness and undecidability
The Löwenheim–Skolem theorem
The compactness theorem
Lindström's theorem


First-order logic
Metalogical properties
Completeness and undecidability


First-order logic
Metalogical properties
The Löwenheim–Skolem theorem


First-order logic
Metalogical properties
The compactness theorem


First-order logic
Metalogical properties
Lindström's theorem


Expressiveness
Formalizing natural languages


First-order logic
Limitations
Expressiveness


First-order logic
Limitations
Formalizing natural languages


Restrictions, extensions, and variations

Restricted languages
Many-sorted logic
Additional quantifiers
Infinitary logics
Non-classical and modal logics
Fixpoint logic
Higher-order logics


Automated theorem proving and formal methods


Vocabulary
Forms of fractions
Arithmetic with fractions
Fractions in abstract mathematics
Algebraic fractions
Radical expressions
Typographical variations
History
In formal education


Reciprocals and the "invisible denominator"


Fraction
Forms of fractions
Simple, common, or vulgar fractions


Fraction
Forms of fractions
Proper and improper fractions


Fraction
Forms of fractions
Decimal fractions and percentages


Fraction
Forms of fractions
Mixed numbers


Fraction
Forms of fractions
Historical notions

Egyptian fraction
Complex and compound fractions


Fraction
Forms of fractions
Historical notions
Egyptian fraction


Fraction
Forms of fractions
Historical notions
Complex and compound fractions


Converting between decimals and fractions


Fraction
Arithmetic with fractions
Equivalent fractions


Fraction
Arithmetic with fractions
Equivalent fractions
Simplifying (reducing) fractions


Fraction
Arithmetic with fractions
Comparing fractions


Fraction
Arithmetic with fractions
Addition


Fraction
Arithmetic with fractions
Addition
Adding unlike quantities


Fraction
Arithmetic with fractions
Subtraction


Fraction
Arithmetic with fractions
Multiplication

Multiplying a fraction by another fraction

Multiplying a fraction by a whole number


Multiplying a fraction by another fraction


Multiplying a fraction by a whole number


Fraction
Arithmetic with fractions
Multiplication
Multiplying mixed numbers


Fraction
Arithmetic with fractions
Division


Converting between decimals and fractions

Converting repeating decimals to fractions


Converting repeating decimals to fractions


Fraction
Fractions in abstract mathematics


Pedagogical tools
Documents for teachers


Fraction
In formal education
Pedagogical tools


Fraction
In formal education
Documents for teachers


Minimal model of set theory is countable


Definition
Notation
Visualization
History
Relations and operations
Properties
Formal construction
Complex analysis
Applications
Generalizations and related notions


Cartesian complex plane
Polar complex plane
Complex graphs


Complex number
Visualization
Cartesian complex plane


Complex number
Visualization
Polar complex plane


Complex number
Visualization
Polar complex plane
Modulus and argument


Complex number
Visualization
Complex graphs


Multiplication and division in polar form


Complex number
Relations and operations
Equality


Complex number
Relations and operations
Ordering


Complex number
Relations and operations
Conjugate


Complex number
Relations and operations
Addition and subtraction


Complex number
Relations and operations
Multiplication and square


Complex number
Relations and operations
Reciprocal and division


Complex number
Relations and operations
Square root


Complex number
Relations and operations
Exponential function


Complex number
Relations and operations
Exponential function
Functional equation


Complex number
Relations and operations
Exponential function
Euler's formula


Complex number
Relations and operations
Complex logarithm


Complex number
Relations and operations
Exponentiation


Complex number
Relations and operations
Exponentiation
Integer and fractional exponents


Field structure
Solutions of polynomial equations
Algebraic characterization
Characterization as a topological field


Complex number
Properties
Field structure


Complex number
Properties
Solutions of polynomial equations


Complex number
Properties
Algebraic characterization


Complex number
Properties
Characterization as a topological field


Matrix representation of complex numbers


Complex number
Formal construction
Construction as ordered pairs


Complex number
Formal construction
Construction as a quotient field


Complex exponential and related functions


Complex number
Complex analysis
Holomorphic functions


Geometry
Algebraic number theory
Analytic number theory
Improper integrals
Dynamic equations
Linear algebra
In applied mathematics
In physics


Complex number
Applications
Geometry
Shapes


Complex number
Applications
Geometry
Fractal geometry


Complex number
Applications
Geometry
Triangles


Complex number
Applications
Algebraic number theory


Complex number
Applications
Analytic number theory


Complex number
Applications
Improper integrals


Complex number
Applications
Dynamic equations


Complex number
Applications
Linear algebra


Complex number
Applications
In applied mathematics


Complex number
Applications
In applied mathematics
Control theory


Complex number
Applications
In applied mathematics
Signal analysis


Electromagnetism and electrical engineering


Complex number
Applications
In physics
Fluid dynamics


Complex number
Applications
In physics
Quantum mechanics


Complex number
Applications
In physics
Relativity


Complex number
Generalizations and related notions


Definition
Properties
⊂ and ⊃ symbols
Examples of subsets
Other properties of inclusion


Mathematics
Physics
Computing
Phonetics and orthography
Literature


Pipe
Disjunction
Concatenation
Delimiter
Backus–Naur form
Concurrency operator
APL
List comprehensions
Text markup


Vertical bar
Usage
Computing
Disjunction


Vertical bar
Usage
Computing
Concatenation


Vertical bar
Usage
Computing
Backus–Naur form


Vertical bar
Usage
Computing
Concurrency operator


Vertical bar
Usage
Computing
List comprehensions


Vertical bar
Usage
Computing
Text markup


Vertical bar
Usage
Phonetics and orthography


Punctuation
Poetry
Notation
Music scoring


Vertical bar
Usage
Literature
Punctuation


Vertical bar
Usage
Literature
Music scoring


Code pages and other historical encodings


Vertical bar
Encoding
Solid vertical bar vs broken bar


Vertical bar
Encoding
Unicode code points


Introduction
Common axioms
Common algebraic structures
Hybrid structures
Universal algebra
Category theory
Different meanings of "structure"


Equational axioms
Existential axioms
Non-equational axioms


Algebraic structure
Common axioms
Equational axioms


Algebraic structure
Common axioms
Existential axioms


Algebraic structure
Common axioms
Non-equational axioms


Algebraic structure
Common algebraic structures

One set with operations
Two sets with operations


Algebraic structure
Common algebraic structures
One set with operations


Algebraic structure
Common algebraic structures
Two sets with operations


Algebraic structure
Different meanings of "structure"


History
Basic concepts
Branches of abstract algebra
Applications


Elementary algebra
Early group theory
Early ring theory
Early field theory
Other major areas
Modern algebra


Abstract algebra
History
Elementary algebra


Abstract algebra
History
Early group theory


Abstract algebra
History
Early ring theory


Abstract algebra
History
Early field theory


Abstract algebra
History
Other major areas


Abstract algebra
Branches of abstract algebra


Abstract algebra
Branches of abstract algebra
Group theory


Abstract algebra
Branches of abstract algebra
Ring theory


Elementary consequences of the group axioms


Group (mathematics)
Definition and illustration

First example: the integers
Definition
Notation and terminology
Second example: a symmetry group


Group (mathematics)
Definition and illustration
First example: the integers


Group (mathematics)
Definition and illustration
Definition


Group (mathematics)
Definition and illustration
Notation and terminology


Group (mathematics)
Definition and illustration
Second example: a symmetry group


Elementary consequences of the group axioms

Equivalent definition with relaxed axioms


Equivalent definition with relaxed axioms


Group homomorphisms
Subgroups
Cosets
Quotient groups
Presentations


Group (mathematics)
Basic concepts
Group homomorphisms


Group (mathematics)
Basic concepts
Subgroups


Group (mathematics)
Basic concepts
Cosets


Group (mathematics)
Basic concepts
Quotient groups


Group (mathematics)
Basic concepts
Presentations


Group (mathematics)
Examples and applications

General linear group and representation theory


Group (mathematics)
Examples and applications
Numbers


Group (mathematics)
Examples and applications
Numbers
Integers


Group (mathematics)
Examples and applications
Numbers
Rationals


Group (mathematics)
Examples and applications
Modular arithmetic


Group (mathematics)
Examples and applications
Cyclic groups


Group (mathematics)
Examples and applications
Symmetry groups


General linear group and representation theory


Group (mathematics)
Examples and applications
Galois groups


Finite abelian groups
Simple groups
Classification of finite simple groups


Group (mathematics)
Finite groups
Finite abelian groups


Group (mathematics)
Finite groups
Simple groups


Group (mathematics)
Finite groups
Classification of finite simple groups


Group (mathematics)
Groups with additional structure


Group (mathematics)
Groups with additional structure
Topological groups


Group (mathematics)
Groups with additional structure
Lie groups


Definition
Examples
Elementary notions
Finite fields
History
Constructing fields
Fields with additional structure
Galois theory
Invariants of fields
Applications
Related notions
Notes
References
External links


Classic definition
Alternative definition


Field (mathematics)
Definition
Classic definition


Field (mathematics)
Definition
Alternative definition


Rational numbers
Real and complex numbers
Constructible numbers
A field with four elements


Field (mathematics)
Examples
Rational numbers


Field (mathematics)
Examples
Real and complex numbers


Field (mathematics)
Examples
Constructible numbers


Field (mathematics)
Examples
A field with four elements


The additive and the multiplicative group of a field


Field (mathematics)
Elementary notions
Consequences of the definition


Field (mathematics)
Elementary notions
Characteristic


Field (mathematics)
Elementary notions
Subfields and prime fields


Constructing fields within a bigger field


Field (mathematics)
Constructing fields
Constructing fields from rings


Field (mathematics)
Constructing fields
Constructing fields from rings
Field of fractions


Field (mathematics)
Constructing fields
Constructing fields from rings
Residue fields


Field (mathematics)
Constructing fields
Field extensions

Algebraic extensions
Transcendence bases


Field (mathematics)
Constructing fields
Field extensions
Algebraic extensions


Field (mathematics)
Constructing fields
Field extensions
Transcendence bases


Field (mathematics)
Constructing fields
Closure operations


Field (mathematics)
Fields with additional structure

Ordered fields
Topological fields
Differential fields


Field (mathematics)
Fields with additional structure
Ordered fields


Field (mathematics)
Fields with additional structure
Topological fields


Field (mathematics)
Fields with additional structure
Topological fields
Local fields


Field (mathematics)
Fields with additional structure
Differential fields


Field (mathematics)
Invariants of fields

Model theory of fields
The absolute Galois group
K-theory


Field (mathematics)
Invariants of fields
Model theory of fields


Field (mathematics)
Invariants of fields
The absolute Galois group


Field (mathematics)
Invariants of fields
K-theory


Finite fields: cryptography and coding theory


Field (mathematics)
Applications
Linear algebra and commutative algebra


Field (mathematics)
Applications
Geometry: field of functions


Field (mathematics)
Applications
Number theory: global fields


Field (mathematics)
Related notions
Division rings


Definition
Illustration
History
Basic examples
Basic concepts
Module
Constructions
Special kinds of rings
Rings with extra structure
Some examples of the ubiquity of rings
Category-theoretic description
Generalization
Other ring-like objects


Ring (mathematics)
Definition
Notes on the definition


Some properties
Example: Integers modulo 4
Example: 2-by-2 matrices


Ring (mathematics)
Illustration
Some properties


Ring (mathematics)
Illustration
Example: Integers modulo 4


Ring (mathematics)
Illustration
Example: 2-by-2 matrices


Multiplicative identity and the term "ring"


Ring (mathematics)
History
Fraenkel and Noether


Commutative rings
Noncommutative rings
Non-rings


Ring (mathematics)
Basic examples
Commutative rings


Ring (mathematics)
Basic examples
Noncommutative rings


Ring (mathematics)
Basic examples
Non-rings


Products and powers
Elements in a ring
Subring
Ideal
Homomorphism
Quotient ring


Ring (mathematics)
Basic concepts
Products and powers


Ring (mathematics)
Basic concepts
Elements in a ring


Ring (mathematics)
Basic concepts
Subring


Ring (mathematics)
Basic concepts
Homomorphism


Ring (mathematics)
Basic concepts
Quotient ring


Direct product
Polynomial ring
Matrix ring and endomorphism ring
Limits and colimits of rings
Localization
Completion
Rings with generators and relations


Ring (mathematics)
Constructions
Direct product


Ring (mathematics)
Constructions
Polynomial ring


Ring (mathematics)
Constructions
Matrix ring and endomorphism ring


Ring (mathematics)
Constructions
Limits and colimits of rings


Ring (mathematics)
Constructions
Localization


Ring (mathematics)
Constructions
Completion


Ring (mathematics)
Constructions
Rings with generators and relations


Ring (mathematics)
Special kinds of rings

Domains
Division ring
Semisimple rings
Central simple algebra and Brauer group
Valuation ring


Ring (mathematics)
Special kinds of rings
Domains


Ring (mathematics)
Special kinds of rings
Division ring


Ring (mathematics)
Special kinds of rings
Semisimple rings


Ring (mathematics)
Special kinds of rings
Semisimple rings
Examples


Ring (mathematics)
Special kinds of rings
Semisimple rings
Properties


Ring (mathematics)
Special kinds of rings
Central simple algebra and Brauer group


Ring (mathematics)
Special kinds of rings
Valuation ring


Ring (mathematics)
Rings with extra structure


Ring (mathematics)
Some examples of the ubiquity of rings

Function field of an irreducible algebraic variety


Ring (mathematics)
Some examples of the ubiquity of rings
Cohomology ring of a topological space


Ring (mathematics)
Some examples of the ubiquity of rings
Burnside ring of a group


Ring (mathematics)
Some examples of the ubiquity of rings
Representation ring of a group ring


Function field of an irreducible algebraic variety


Ring (mathematics)
Some examples of the ubiquity of rings
Face ring of a simplicial complex


Ring (mathematics)
Category-theoretic description


Ring (mathematics)
Generalization
Nonassociative ring


Ring (mathematics)
Generalization
Semiring


Ring (mathematics)
Other ring-like objects

Ring object in a category
Ring scheme
Ring spectrum


Ring (mathematics)
Other ring-like objects
Ring object in a category


Ring (mathematics)
Other ring-like objects
Ring scheme


Ring (mathematics)
Other ring-like objects
Ring spectrum


Definitions
In topology
Binary relations
Other examples
Closure operator
Notes
References


Closure (mathematics)
Definitions
In algebraic structures


Closure (mathematics)
Closure operator
Closure operator vs. closed sets


Natural domain
Other uses
Set theoretical notions


Domain of a function
Natural domain
Examples


Domain of a function
Set theoretical notions


Early life and background
Childhood and adolescence
University and first marriage
Early career
First World War
G. H. Hardy on the Trinity controversy
Between the wars
Second World War
Later life
Political causes
Final years, death and legacy
Titles and honours from birth


Bertrand Russell
Biography
Early life and background


Bertrand Russell
Biography
Childhood and adolescence


Bertrand Russell
Biography
University and first marriage


Bertrand Russell
Biography
First World War


Bertrand Russell
Biography
G. H. Hardy on the Trinity controversy


Bertrand Russell
Biography
Between the wars


Bertrand Russell
Biography
Second World War


Bertrand Russell
Biography
Political causes


Bertrand Russell
Biography
Final years, death and legacy


Bertrand Russell
Biography
Final years, death and legacy
Marriages and issue


Bertrand Russell
Biography
Titles and honours from birth


Philosophy
Religion
Society
Freedom of opinion and expression
Education


Bertrand Russell
Views
Freedom of opinion and expression


Etymology
Gameplay
History
Strategies
Variations
Analogues in real life
Instances of usage
Tournaments


Rock paper scissors
History
Spread beyond East Asia


Rock paper scissors
Strategies
Algorithms


Adapted rules
Different weapons
Additional weapons


Rock paper scissors
Variations
Adapted rules


Rock paper scissors
Variations
Different weapons


Rock paper scissors
Variations
Additional weapons


Rock paper scissors
Analogues in real life

Mechanical devices and geometrical constructions


Rock paper scissors
Analogues in real life
Lizard mating strategies


Rock paper scissors
Analogues in real life
Bacteria


Mechanical devices and geometrical constructions


American court case
Auction house selection
FA Women's Super League match
Play by chimpanzees
Game design


Rock paper scissors
Instances of usage
American court case


Rock paper scissors
Instances of usage
Auction house selection


Rock paper scissors
Instances of usage
FA Women's Super League match


Rock paper scissors
Instances of usage
Play by chimpanzees


Rock paper scissors
Instances of usage
Game design


National XtremeRPS Competition 2007–2008


Rock paper scissors
Tournaments
World Rock Paper Scissors Association


Rock paper scissors
Tournaments
World Rock Paper Scissors Society


Rock paper scissors
Tournaments
UK championships


Rock paper scissors
Tournaments
USARPS tournaments


Rock paper scissors
Tournaments
Guinness Book of World Records


Rock paper scissors
Tournaments
World Series


Rock paper scissors
Tournaments
Jackpot En Poy of Eat Bulaga!


Vopěnka's Alternative Set Theory
Other alternative set theories


Alternative set theory
Vopěnka's Alternative Set Theory


Alternative set theory
Other alternative set theories


Properties of the category of sets
Foundations for the category of sets


Category of sets
Properties of the category of sets


Category of sets
Foundations for the category of sets


Examples
Paradoxes
Classes in formal set theories
Notes
References
External links


Class (set theory)
Classes in formal set theories


Formula
Examples
A special case
Formula generalization
In probability
Other formulas
Applications
Diluted inclusion–exclusion principle
Proof of main statement


Inclusion–exclusion principle
Examples
Counting integers


Inclusion–exclusion principle
Examples
Counting derangements


Inclusion–exclusion principle
A special case


Inclusion–exclusion principle
Formula generalization


Inclusion–exclusion principle
In probability


Inclusion–exclusion principle
In probability
Special case


Inclusion–exclusion principle
Other formulas


Counting derangements
Counting intersections
Graph coloring
Bipartite graph perfect matchings
Number of onto functions
Permutations with forbidden positions
Stirling numbers of the second kind
Rook polynomials
Euler's phi function


Inclusion–exclusion principle
Applications
Counting derangements


Inclusion–exclusion principle
Applications
Counting intersections


Inclusion–exclusion principle
Applications
Graph coloring


Inclusion–exclusion principle
Applications
Bipartite graph perfect matchings


Inclusion–exclusion principle
Applications
Number of onto functions


Inclusion–exclusion principle
Applications
Permutations with forbidden positions


Inclusion–exclusion principle
Applications
Stirling numbers of the second kind


Inclusion–exclusion principle
Applications
Rook polynomials


Inclusion–exclusion principle
Applications
Euler's phi function


Inclusion–exclusion principle
Diluted inclusion–exclusion principle


Inclusion–exclusion principle
Proof of main statement


Inclusion–exclusion principle
Proof of main statement
Algebraic proof


Definition
Examples
Properties
Related notions


Dense set
Definition
Density in metric spaces


Examples
Properties
Related concepts
Special types of set families


Family of sets
Related concepts
Covers and topologies


Family of sets
Special types of set families


Definition
Fuzzy logic
Fuzzy number and only number
Fuzzy categories
Fuzzy relation equation
Entropy
Extensions


Crisp sets related to a fuzzy set
Other definitions
Fuzzy set operations
Disjoint fuzzy sets
Scalar cardinality
Distance and similarity
L-fuzzy sets
Neutrosophic fuzzy sets
Pythagorean fuzzy sets


Fuzzy set
Definition
Crisp sets related to a fuzzy set


Fuzzy set
Definition
Fuzzy set operations


Fuzzy set
Definition
Disjoint fuzzy sets


Fuzzy set
Definition
Distance and similarity


Fuzzy set
Definition
Neutrosophic fuzzy sets


Fuzzy set
Definition
Pythagorean fuzzy sets


Internal sets in the ultrapower construction


Internal set
Internal subsets of the reals


History
Axioms and primitive notions
Various systems
Set theory
Mathematics
Natural language
Metaphysics
Important surveys


Mereological constitution
Mereological composition


Mereology
Metaphysics
Mereological constitution


Mereology
Metaphysics
Mereological composition

Fundamentality
Special composition question (SCQ)


Mereology
Metaphysics
Mereological composition
Fundamentality


Mereology
Metaphysics
Mereological composition
Special composition question (SCQ)


History
Examples
Definition
Basic properties and operations
Counting multisets
Applications
Generalizations


Multiset
Basic properties and operations


Generalization and connection to the negative binomial series


Multiset
Counting multisets
Recurrence relation


Multiset
Counting multisets
Generating series


Ramified types and the axiom of reducibility


Principia Mathematica
Scope of foundations laid


Contemporary construction of a formal theory


Principia Mathematica
Theoretical basis
Construction


Principia Mathematica
Theoretical basis
Primitive ideas


Principia Mathematica
Theoretical basis
Primitive propositions


An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)

An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)

Introduction to the notation of the theory of classes and relations


An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)


An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)


Introduction to the notation of the theory of classes and relations


Principia Mathematica
Consistency and criticisms

Gödel 1930, 1931
Wittgenstein 1919, 1939
Gödel 1944


Principia Mathematica
Consistency and criticisms
Gödel 1930, 1931


Principia Mathematica
Consistency and criticisms
Wittgenstein 1919, 1939


Principia Mathematica
Consistency and criticisms
Gödel 1944


Part I Mathematical logic. Volume I ✸1 to ✸43

Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97

Part III Cardinal arithmetic. Volume II ✸100 to ✸126

Part IV Relation-arithmetic. Volume II ✸150 to ✸186

Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276

Part VI Quantity. Volume III ✸300 to ✸375


Part I Mathematical logic. Volume I ✸1 to ✸43


Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97


Part III Cardinal arithmetic. Volume II ✸100 to ✸126


Part IV Relation-arithmetic. Volume II ✸150 to ✸186


Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276


Part VI Quantity. Volume III ✸300 to ✸375


Principia Mathematica
Comparison with set theory


Principia Mathematica
Differences between editions


Role and characterizations in mathematics


Name
Definition
Irrationality and normality
Transcendence
Continued fractions
Approximate value and digits
Complex numbers and Euler's identity


Pi
Fundamentals
Irrationality and normality


Pi
Fundamentals
Approximate value and digits


Pi
Fundamentals
Complex numbers and Euler's identity


Antiquity
Polygon approximation era
Infinite series
Irrationality and transcendence
Adoption of the symbol π


Pi
History
Infinite series
Rate of convergence


Pi
History
Irrationality and transcendence


Computer era and iterative algorithms
Motives for computing π
Rapidly convergent series
Monte Carlo methods
Spigot algorithms


Pi
Modern quest for more digits
Computer era and iterative algorithms


Pi
Modern quest for more digits
Motives for computing π


Pi
Modern quest for more digits
Rapidly convergent series


Pi
Modern quest for more digits
Monte Carlo methods


Pi
Modern quest for more digits
Spigot algorithms


Role and characterizations in mathematics

Fourier transform and Heisenberg uncertainty principle

The gamma function and Stirling's approximation

Cauchy distribution and potential theory


Fourier transform and Heisenberg uncertainty principle


The gamma function and Stirling's approximation


Cauchy distribution and potential theory


Describing physical phenomena
Memorizing digits
In popular culture
In computer culture


Pi
Outside mathematics
Describing physical phenomena


Pi
Outside mathematics
Memorizing digits


Pi
Outside mathematics
In popular culture


Pi
Outside mathematics
In computer culture


Definitions
Rule extraction
Incomplete data
Applications
History
Extensions and generalizations


Information system framework
Example: equivalence-class structure
Definition of a rough set
Definability
Reduct and core
Attribute dependency


Rough set
Definitions
Information system framework


Rough set
Definitions
Example: equivalence-class structure


Rough set
Definitions
Definition of a rough set

Lower approximation and positive region
Upper approximation and negative region
Boundary region
The rough set
Objective analysis


Rough set
Definitions
Definition of a rough set
Lower approximation and positive region


Rough set
Definitions
Definition of a rough set
Upper approximation and negative region


Rough set
Definitions
Definition of a rough set
Boundary region


Rough set
Definitions
Definition of a rough set
The rough set


Rough set
Definitions
Definition of a rough set
Objective analysis


Rough set
Definitions
Attribute dependency


Decision matrices
LERS rule induction system


Rough set
Rule extraction
Decision matrices


Rough set
Rule extraction
LERS rule induction system


Rough set
Extensions and generalizations


Rough set
Extensions and generalizations
Rough membership


Rough set
Extensions and generalizations
Other generalizations


EAN format used in barcodes, and upgrading


Issuing process
Registration group element
Registrant element


ISBN
Overview
Registration group element


ISBN
Overview
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Contents
Early reviews
Second edition
Later reviews


The Principles of Mathematics
Early reviews


The Principles of Mathematics
Second edition


The Principles of Mathematics
Later reviews


History
Past editors
References
External links


Comparison with other identifier schemes


Digital object identifier
Nomenclature and syntax


Digital object identifier
Nomenclature and syntax
Display


Digital object identifier
Features and benefits


Digital object identifier
IDF organizational structure


Digital object identifier
Standardization


Aaron Swartz incident
Limitations
Increasing public access


Website design
Search
Journal article parameters


PubMed
Characteristics
Website design
PubMed for handhelds/mobiles


PubMed
Characteristics
Search
Standard search


PubMed
Characteristics
Search
Comprehensive search


PubMed
Characteristics
Journal article parameters

Publication Type: Clinical queries/systematic reviews


Publication Type: Clinical queries/systematic reviews


PubMed
Characteristics
Journal article parameters
Secondary ID


Positions
Publications
Articles, reviews, and essays


Joseph Dauben
Articles, reviews, and essays


Classic reprints
History
Publications
References
External links


Induced substructures and closed subsets


Structure (mathematical logic)
Definition

Domain
Signature
Interpretation function
Examples


Structure (mathematical logic)
Definition
Domain


Structure (mathematical logic)
Definition
Signature


Structure (mathematical logic)
Definition
Interpretation function


Structure (mathematical logic)
Definition
Examples


Structure (mathematical logic)
Homomorphisms and embeddings

Homomorphisms
Embeddings
Example
Homomorphism problem


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphisms


Structure (mathematical logic)
Homomorphisms and embeddings
Embeddings


Structure (mathematical logic)
Homomorphisms and embeddings
Example


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphism problem


Structure (mathematical logic)
Structures and first-order logic

Satisfaction relation
Definable relations


Structure (mathematical logic)
Structures and first-order logic
Satisfaction relation


Structure (mathematical logic)
Structures and first-order logic
Definable relations

Definability with parameters
Implicit definability


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Definability with parameters


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Implicit definability


Structure (mathematical logic)
Many-sorted structures


Structure (mathematical logic)
Other generalizations

Partial algebras
Structures for typed languages
Higher-order languages
Structures that are proper classes


Structure (mathematical logic)
Other generalizations
Partial algebras


Structure (mathematical logic)
Other generalizations
Structures for typed languages


Structure (mathematical logic)
Other generalizations
Higher-order languages


Structure (mathematical logic)
Other generalizations
Structures that are proper classes


A priori property of logical consequence


Syntactic consequence
Semantic consequence


Logical consequence
Proofs and models
Syntactic consequence


Logical consequence
Proofs and models
Semantic consequence


Modal-formal accounts
Warrant-based accounts
Non-monotonic logical consequence


Logical consequence
Modal accounts
Modal-formal accounts


Logical consequence
Modal accounts
Warrant-based accounts


Logical consequence
Modal accounts
Non-monotonic logical consequence


Nobel prize winners published by Cambridge University Press

Organisational governance and operational structure

Cambridge University Press partnerships and acquisitions


Cambridge University Press
Print and typographic heritage


Cambridge University Press
Print and typographic heritage
People


Cambridge University Press
Print and typographic heritage
Publications


Cambridge University Press
Current publications


Nobel prize winners published by Cambridge University Press


Organisational governance and operational structure

Relationship with the University of Cambridge


Relationship with the University of Cambridge


Cambridge University Press partnerships and acquisitions


Cambridge University Press
Digital developments


Cambridge University Press
Controversies

Tax exemption controversy
Alms for Jihad
Cambridge University Press v. Patton
The China Quarterly
The Cambridge Handbook of Privatization


Cambridge University Press
Controversies
Tax exemption controversy


Cambridge University Press
Controversies
Alms for Jihad


Cambridge University Press
Controversies
Cambridge University Press v. Patton


Cambridge University Press
Controversies
The China Quarterly


Cambridge University Press
Controversies
The Cambridge Handbook of Privatization


Cambridge University Press
Corporate social responsibility


Cambridge University Press
Corporate social responsibility
Community


Cambridge University Press
Corporate social responsibility
Environment


Cambridge University Press
References
Citations


Cambridge University Press
References
Sources


Cambridge University Press
External links


Formal language
Formal grammar
Formal systems
Interpretations


Etymology
Historical development
Mathematical logic


Early Greeks
Modern development
Other sciences


Axiom
Historical development
Early Greeks


Axiom
Historical development
Modern development


Axiom
Historical development
Other sciences


Logical axioms
Non-logical axioms
Role in mathematical logic
Further discussion


Axiom
Mathematical logic
Logical axioms
Examples


Axiom
Mathematical logic
Non-logical axioms


Axiom
Mathematical logic
Non-logical axioms
Examples


Axiom
Mathematical logic
Role in mathematical logic


Axiom
Mathematical logic
Role in mathematical logic
Deductive systems and completeness


Axiom
Mathematical logic
Further discussion


Related publishers, imprints, and series


Early life and education
Career
Accomplishments
Books by Halmos


Proof theory and constructive mathematics


Mathematical logic
History
Early history


Mathematical logic
History
19th century
Foundational theories


Set theory and paradoxes
Symbolic logic
Beginnings of the other branches


Mathematical logic
History
20th century
Set theory and paradoxes


Mathematical logic
History
20th century
Symbolic logic


Mathematical logic
History
20th century
Beginnings of the other branches


Mathematical logic
Formal logical systems

First-order logic
Other classical logics
Nonclassical and modal logic
Algebraic logic


Mathematical logic
Formal logical systems
First-order logic


Mathematical logic
Formal logical systems
Other classical logics


Mathematical logic
Formal logical systems
Nonclassical and modal logic


Mathematical logic
Formal logical systems
Algebraic logic


Mathematical logic
Recursion theory
Algorithmically unsolvable problems


Mathematical logic
Connections with computer science


Mathematical logic
Foundations of mathematics


History
Introduction
Differences from set theory
Technical details
Interpretations
List of type theories
Applications


Basics
Functions
Common "by rule" types and terms


Type theory
Introduction
Basics
Terms and types


Type theory
Introduction
Basics
Computation


Type theory
Introduction
Functions
Lambda terms


Type theory
Introduction
Functions
Dependent typing


Type theory
Introduction
Functions
Universes


Type theory
Introduction
Common "by rule" types and terms

Empty type
Unit type
Boolean type
Product type
Sum type
Natural numbers
Identity type
Inductive types


Type theory
Introduction
Common "by rule" types and terms
Empty type


Type theory
Introduction
Common "by rule" types and terms
Unit type


Type theory
Introduction
Common "by rule" types and terms
Boolean type


Type theory
Introduction
Common "by rule" types and terms
Product type


Type theory
Introduction
Common "by rule" types and terms
Sum type


Type theory
Introduction
Common "by rule" types and terms
Natural numbers


Type theory
Introduction
Common "by rule" types and terms
Identity type


Type theory
Introduction
Common "by rule" types and terms
Inductive types


Terms
Judgements
Rules
Properties of type theories
Axioms
Decision problems
Research areas


Type theory
Technical details
Judgements


Type theory
Technical details
Properties of type theories


Type theory
Technical details
Decision problems


Type theory
Technical details
Decision problems
Type inhabitation


Type theory
Technical details
Decision problems
Type inference


Type theory
Technical details
Research areas


Types are propositions; terms are proofs


Type theory
Interpretations
Curry-Howard correspondence


Type theory
Interpretations
Intuitionistic logic


Type theory
Interpretations
Constructive mathematics


Type theory
Interpretations
Category theory


Type theory
List of type theories
Active research


Mathematical foundations
Proof assistants
Programming languages
Linguistics
Social sciences


Type theory
Applications
Mathematical foundations


Type theory
Applications
Proof assistants


Type theory
Applications
Programming languages


Type theory
Applications
Social sciences


Theoremhood and truth
Epistemological considerations
Informal account of theorems
Relation with scientific theories
Terminology
Layout
Lore
Theorems in logic


Theorem
Relation with scientific theories


Syntax and semantics
Interpretation of a formal theorem
Theorems and theories


Theorem
Theorems in logic
Syntax and semantics


Theorem
Theorems in logic
Interpretation of a formal theorem


Theorem
Theorems in logic
Theorems and theories


Basics
Paradoxes of the infinite sets
Paradoxes of the Supertask
Paradoxes of proof and definability


Cardinal numbers
Ordinal numbers
Power sets


Paradoxes of set theory
Basics
Cardinal numbers


Paradoxes of set theory
Basics
Ordinal numbers


Paradoxes of set theory
Basics
Power sets


Paradoxes of set theory
Paradoxes of the infinite sets

Paradoxes of enumeration
Je le vois, mais je ne crois pas
Paradoxes of well-ordering


Paradoxes of set theory
Paradoxes of the infinite sets
Paradoxes of enumeration


Paradoxes of set theory
Paradoxes of the infinite sets
Je le vois, mais je ne crois pas


Paradoxes of set theory
Paradoxes of the infinite sets
Paradoxes of well-ordering


Paradoxes of set theory
Paradoxes of the Supertask

The diary of Tristram Shandy
The Ross-Littlewood paradox


Paradoxes of set theory
Paradoxes of the Supertask
The diary of Tristram Shandy


Paradoxes of set theory
Paradoxes of the Supertask
The Ross-Littlewood paradox


Paradoxes of set theory
Paradoxes of proof and definability

Early paradoxes: the set of all sets
Paradoxes by change of language
Paradox of Löwenheim and Skolem


Paradoxes of set theory
Paradoxes of proof and definability
Early paradoxes: the set of all sets


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language
König's paradox


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language
Richard's paradox


Paradoxes of set theory
Paradoxes of proof and definability
Paradox of Löwenheim and Skolem


Etymology
Comparison with theorem
Well-known lemmas


Lemma (mathematics)
Comparison with theorem


ZF (the Zermelo–Fraenkel axioms without the axiom of choice)