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Set of the elements not in a given subset
In set theory, the complement of a set A, often denoted by Ac (or A′), is the set of elements not in A. When all elements in the universe, i.e. all elements
Complement_(set_theory)
Identities and relationships involving sets
of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and
Algebra_of_sets
Set of elements in any of some sets
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations
Union_(set_theory)
Set of elements common to all of some sets
In set theory, the intersection of two sets A and B, denoted by A∩B, is the set containing all elements of A that also belong to B or equivalently, all
Intersection_(set_theory)
Elements in exactly one of two sets
of sets Boolean function Complement (set theory) Difference (set theory) Exclusive or Fuzzy set Intersection (set theory) Jaccard index List of set identities
Symmetric_difference
Class (set theory) Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive
List_of_set_theory_topics
Branch of music theory
inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well. Although musical set theory is often
Set_theory_(music)
Standard system of axiomatic set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in
Zermelo–Fraenkel_set_theory
Branch of mathematics that studies sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of unique objects. Although objects
Set_theory
Topics referred to by the same term
(sometimes called an antonym) Complement (group theory) Complementary subspaces Orthogonal complement Schur complement Complement (complexity), relating to
Complement
Set whose pairs have minima and maxima
the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique
Lattice_(order)
Graph with same nodes as but complementary connections to another
In the mathematical field of graph theory, the complement or inverse of a graph G is a graph H on the same vertices such that two distinct vertices are
Complement_graph
Subfield of mathematical logic
Borel set is Borel, not all analytic sets are Borel sets. A set is coanalytic if its complement is analytic. Many questions in descriptive set theory ultimately
Descriptive_set_theory
Informal set theories
Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined
Naive_set_theory
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable, or recursive) if there is an algorithm that computes the membership of
Computable_set
Any one of the distinct objects that make up a set in set theory
"Set Theory", Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University Suppes, Patrick (1972) [1960], Axiomatic Set Theory,
Element_of_a_set
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that G = H
Complement_(group_theory)
Sequence of words formed by specific rules
computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages
Formal_language
Bound lattice in which every element has a complement
order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e
Complemented_lattice
Unrelated vertices in graphs
graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S
Independent set (graph theory)
Independent_set_(graph_theory)
Concept in music
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism. In interval
Complement_(music)
Set whose elements all belong to another set
k-subsets of an n-element set. In set theory, the notation [A]k is also common, especially when k is a transfinite cardinal number. A set A is a subset of B
Subset
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Mathematical set containing no elements
empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure
Empty_set
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
All-encompassing set or class
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains
Universe_(mathematics)
Collection of sets in mathematics that can be defined based on a property of its members
In set theory and its applications throughout mathematics, a class is a collection of mathematical objects (often sets) that can be unambiguously defined
Class_(set_theory)
System of mathematical set theory
mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine
Morse–Kelley_set_theory
descriptive set theory, a scale is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of
Scale (descriptive set theory)
Scale_(descriptive_set_theory)
System of mathematical set theory
set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF)
Zermelo_set_theory
Equalities for combinations of sets
and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion
List of set identities and relations
List_of_set_identities_and_relations
In computability theory, a maximal set is a coinfinite computably enumerable subset A of the natural numbers such that for every further computably enumerable
Maximal set (computability theory)
Maximal_set_(computability_theory)
Axiomatic set theories based on the principles of mathematical constructivism
Axiomatic constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language
Constructive_set_theory
System of mathematical set theory
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
System of mathematical set theory
Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Mathematical set of all subsets of a set
mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed
Power_set
Any collection of sets, or subsets of a set
In set theory and related branches of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used
Family_of_sets
Class (set theory) Complement (set theory) Complete Boolean algebra Continuum (set theory) Suslin's problem Continuum hypothesis Countable set Descriptive
List of mathematical logic topics
List_of_mathematical_logic_topics
This is a list of set classes, by Forte number. In music theory, a set class (an abbreviation of pitch-class-set class) is an ascending collection of pitch
List_of_set_classes
Binary representation for signed numbers
Two's complement is the most common method of representing signed (positive, negative, and zero) integers on computers, and more generally, fixed point
Two's_complement
Proposition in mathematical logic
specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: There is no set whose
Continuum_hypothesis
Concept in axiomatic set theory
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom)
Axiom_schema_of_specification
Collection of mathematical objects
ultimately defined in terms of sets. That is, set theory serves as foundation of mathematics. Before the end of the 19th century, sets were not studied specifically
Set_(mathematics)
Theorem in set theory
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, κ i {\displaystyle \kappa _{i}} and λ i {\displaystyle \lambda _{i}}
Kőnig's_theorem_(set_theory)
Overview of and topical guide to logic
number Codomain Complement (set theory) Constructible universe Continuum hypothesis Countable set Decidable set Denumerable set Disjoint sets Disjoint union
Outline_of_logic
Algebraic structure of set algebra
mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and
Σ-algebra
In computational complexity theory, the complement of a decision problem is the decision problem resulting from reversing the yes and no answers. Equivalently
Complement_(complexity)
Set theory concept
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the class of hereditary
Von_Neumann_universe
System of mathematical set theory
set theory (KP) is an axiomatic set theory developed by Saul Kripke and Richard Platek. It is substantially weaker than Zermelo–Fraenkel set theory (ZF);
Kripke–Platek_set_theory
Sets whose elements have degrees of membership
does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with
Fuzzy_set
Maximal proper filter
In the mathematical field of set theory, an ultrafilter on a set X {\displaystyle X} is a maximal filter on the set X . {\displaystyle X.} In other words
Ultrafilter_on_a_set
Family closed under unions and relative complements
measure theory, a nonempty family of sets R {\displaystyle {\mathcal {R}}} is called a ring (of sets) if it is closed under union and relative complement (set-theoretic
Ring_of_sets
discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
Size of a possibly infinite set
studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra
Cardinal_number
Family of subsets representing "large" sets
topology, including set theory, mathematical logic, model theory (ultraproducts for example), abstract algebra, and others. Filters on a set were later generalized
Filter_on_a_set
Algebraic concept in measure theory, also referred to as an algebra of sets
Probability theory – Branch of mathematics concerning probability Ring of sets – Family closed under unions and relative complements Set function – Function
Field_of_sets
3-volume treatise on mathematics, 1910–1913
and set theory at the turn of the 20th century, like Russell's paradox. This third aim motivated the adoption of the theory of types in PM. The theory of
Principia_Mathematica
Algebraic manipulation of "true" and "false"
or field of sets is any nonempty set of subsets of a given set X closed under the set operations of union, intersection, and complement relative to X
Boolean_algebra
Branch of mathematics
directed subsets and that are studied in domain theory. Partial orders with complements, or poc sets, are posets with a unique bottom element 0, as well
Order_theory
Theory that allows sets to be elements of themselves
Non-well-founded set theories (sometimes unhyphenated, as nonwellfounded; or poorly founded) are variants of axiomatic set theory that allow sets to be elements
Non-well-founded_set_theory
Axiom used in set theory
in many forms of axiomatic set theory, such as the Zermelo–Fraenkel set theory. Informally, the axiom means that the two sets A and B are equal if and only
Axiom_of_extensionality
Mathematical set containing all objects
In set theory, a universal set is a set that contains all of the objects in the theory, including itself. In set theory as usually formulated, it can be
Universal_set
One-to-one correspondence
This topic is a basic concept in set theory and can be found in any text which includes an introduction to set theory. Almost all texts that deal with
Bijection
Infinite cardinal number
particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced
Aleph_number
Set that is not a finite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. The set of natural numbers (whose existence
Infinite_set
Class of mathematical sets
that contains both the empty set and the entire set X {\displaystyle X} , and is closed under countable union and complement. Then we can define the Borel
Borel_set
Non-contradiction of a theory
formal deductive system. The set of axioms A is consistent when there is no formula φ such that φ∈⟨A⟩ and ¬φ∈⟨A⟩. A trivial theory (i.e., one which proves
Consistency
Set of all things that may be the input of a mathematical function
1971). Axiomatic Set Theory, Part 1. American Mathematical Soc. ISBN 978-0-8218-0245-8. Sharma, A. K. (2010). Introduction To Set Theory. Discovery Publishing
Domain_of_a_function
Appendix:Glossary of set theory in Wiktionary, the free dictionary. This is a glossary of terms and definitions related to the topic of set theory. Contents:
Glossary_of_set_theory
Set with exactly one element
0} . Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton
Singleton_(mathematics)
Type of infinite structure
underlying set. O-minimal structures originated in model theory and so have a simpler—but equivalent—definition using the language of model theory. Namely
O-minimal_theory
computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is infinite)
Simple_set
Axioms for the natural numbers
set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory
Peano_axioms
Ordered listing of items in collection
have little to do with the underlying set, it is useful when some order of the set is necessary. In set theory, there is a more general notion of an enumeration
Enumeration
Term in set theory
set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set
Almost
Axiom of Zermelo-Fraenkel set theory
axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It
Axiom_of_infinity
Pair of logical equivalences
when doing a substitution. In set theory, it is often stated as "union and intersection interchange under complementation", which can be formally expressed
De_Morgan's_laws
Branch of mathematical logic
pp. 3–4), proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1977)
Proof_theory
Type of infinite number in set theory
In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly
Inaccessible_cardinal
German logician and mathematician (1871–1953)
mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work
Ernst_Zermelo
Complexity class used to classify decision problems
computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems
NP_(complexity)
contradictions within modern axiomatic set theory. Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be
Paradoxes_of_set_theory
Paradox in set theory
Russell's paradox. The term "naive set theory" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a first-order
Russell's_paradox
Basic set identities like commutative and associative laws
algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix
Simple theorems in the algebra of sets
Simple_theorems_in_the_algebra_of_sets
Fundamental theorem in mathematical logic
logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model
Gödel's_completeness_theorem
German-Israeli mathematician and Zionist (1891–1965)
contributions to axiomatic set theory, especially his additions to Ernst Zermelo's axioms, which resulted in the Zermelo–Fraenkel set theory. Abraham Adolf Halevi
Abraham_Fraenkel
Mathematical theory of data types
to set theory as a foundation of mathematics. Examples include Alonzo Church's simple theory of types and Per Martin-Löf's intuitionistic type theory. Many
Type_theory
Technique for proving independence results
In set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand
Forcing_(mathematics)
In computability theory, productive sets and creative sets are types of sets of natural numbers that have important applications in mathematical logic
Creative_and_productive_sets
Concept in set theory
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object that is not a set, but that
Urelement
Logical operation
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition P {\displaystyle P} to another proposition
Negation
Set-theoretic concept
In set theory, a Grothendieck universe is a set U {\displaystyle U} with the following properties: If x {\displaystyle x} is an element of U {\displaystyle
Grothendieck_universe
Part of speech
grammatical theories. In traditional grammar, such words are normally considered conjunctions. The standard abbreviation for complementizer is C. The complementizer
Complementizer
Basic framework of mathematics
mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, parts of computer
Foundations_of_mathematics
Axiom of set theory
an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, one can identify another set containing one
Axiom_of_choice
Theorem in set theory
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there
Schröder–Bernstein_theorem
In logic, a statement which is always true
formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i
Tautology_(logic)
Theorem in mathematical logic
a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as
Compactness_theorem
Mathematical concept
is sufficient. Because there are models of Zermelo–Fraenkel set theory of interest to set theorists that satisfy the axiom of dependent choice but not
Transfinite_induction
Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere
List_of_types_of_sets
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COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
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