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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 A c {\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the
Complement_(set_theory)
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)
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 common to all of some sets
In set theory, the intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B , {\displaystyle A\cap B,} is the set containing
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
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
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)
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 objects. Although objects of any
Set_theory
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)
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
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
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
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
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)
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
Set whose elements all belong to another set
of k {\displaystyle k} -subsets of an n {\displaystyle n} -element set. In set theory, the notation [ A ] k {\displaystyle [A]^{k}} is also common, especially
Subset
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
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)
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
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
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
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
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
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
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
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)
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
computable set R contains either only finitely many elements of the complement of A or almost all elements of the complement of A. There are r-maximal sets that
Maximal set (computability theory)
Maximal_set_(computability_theory)
Unrelated vertices in graphs
Ramsey theory. A set is independent if and only if its complement is a vertex cover. Therefore, the sum of the size of the largest independent set α ( G
Independent set (graph theory)
Independent_set_(graph_theory)
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
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 every
Computable_set
Collection of mathematical objects
of sets. Set theory studies possible axiom systems and their consequences. Since the first half of the 20th century, ZFC (Zermelo–Fraenkel set theory with
Set_(mathematics)
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
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
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)
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
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
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
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
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
System of mathematical set theory
General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring
General_set_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
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
System of mathematical set theory
Kripke–Platek set theory (KP), pronounced /ˈkrɪpki ˈplɑːtɛk/, is an axiomatic set theory developed by Saul Kripke and Richard Platek. The theory can be thought
Kripke–Platek_set_theory
Family of subsets representing "large" sets
applications in model theory and set theory. Filters on a set were later generalized to order filters. Specifically, a filter on a set X {\displaystyle X}
Filter_on_a_set
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
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
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
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
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
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
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)
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
Alternative to the standard Zermelo–Fraenkel set theory
Internal set theory Pocket set theory Naive set theory S (set theory) Double extension set theory Kripke–Platek set theory Kripke–Platek set theory with urelements
List of alternative set theories
List_of_alternative_set_theories
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
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
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 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
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
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
Diagram that shows all possible logical relations between a collection of sets
between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships
Venn_diagram
Logical connective AND
languages, the short-circuit and control structure; In set theory, intersection. In lattice theory, logical conjunction (greatest lower bound). And is usually
Logical_conjunction
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)
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
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
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
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
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)
Concept in linear algebra
orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped with a bilinear form B {\displaystyle B} is the set W ⊥
Orthogonal_complement
Size of a set in mathematics
unprovable and undisprovable in standard set theories such as Zermelo–Fraenkel set theory. Alternative set theories and additional axioms give rise to different
Cardinality
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)
concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
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
Study of computable functions and Turing degrees
computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include: What
Computability_theory
Mathematician (1845–1918)
mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance
Georg_Cantor
Mathematical ways to group elements of a set
is sometimes called a setoid, typically in type theory and proof theory. A partition of a set X is a set of non-empty subsets of X such that every element
Partition_of_a_set
Area of mathematical logic
the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes
Model_theory
Method of subtraction
concept of the radix complement (as described below) is also valuable in number theory, such as in Midy's theorem. The nines' complement of a number given
Method_of_complements
Branch of logic
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax)
Finite_model_theory
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
Part of speech
grammatical theories. In traditional grammar, such words are normally considered conjunctions. The standard abbreviation for complementizer is C. The complementizer
Complementizer
Mathematical proposition equivalent to the axiom of choice
as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is,
Zorn's_lemma
Structure of a formal language
such parsers, formal language theory uses separate formalisms, known as automata theory. One result of automata theory is that it is not possible to design
Formal_grammar
Mathematical set with an ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The
Partially_ordered_set
Linguistics theory about syntax
Complementizers: Toward a Syntactic Theory of Complement Types". Foundations of Language. 6: 297–321. Bresnan, Joan (1972) Theory of Complementation in
X-bar_theory
Form of logic that allows quantification over predicates
Zermelo–Fraenkel set theory), as sets are vital for mathematics. Arithmetic, mereology, and a variety of other powerful logical theories could be formulated
Second-order_logic
Logical principle
arose several important logical developments; Zermelo's axiomatization of set theory (1908a), that was followed two years later by the first volume of Principia
Law_of_excluded_middle
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
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
Class of mathematical set whose elements are all subsets
In set theory, a branch of mathematics, a set A {\displaystyle A} is called transitive if either of the following equivalent conditions holds: whenever
Transitive_set
In mathematics, a statement that has been proven
almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an
Theorem
−2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory. × (multiplication sign) 1. In elementary arithmetic
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
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
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
Axiom set used in first-order logic
identity (i.e. is formulable as an elementary theory). As such, it does not require an underlying set theory. The only primitive objects of the system are
Tarski's_axioms
Axiom used in set theory
axiomatic set theory, such as the Zermelo–Fraenkel set theory. The axiom defines what a set is. Informally, the axiom means that the two sets A and B are
Axiom_of_extensionality
Measure of algorithmic complexity
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is
Kolmogorov_complexity
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
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
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
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
Boy/Male
Hindi
Competent.
Male
English
Short form of English Stephen, STE means "crown."
Male
English
Anglicized form of Hebrew Sheth, SETH means "buttocks." In the bible, this is the name of the third son of Adam and Eve. Compare with other forms of Seth.
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Boy/Male
Muslim
Compliments, Happiness
Boy/Male
Arabic, Muslim
Competent
Boy/Male
Japanese
Complacent; satisfied.
Girl/Female
Indian
Competent.
Boy/Male
Arabic, Muslim
Competent
Boy/Male
Arabic, Muslim
Competent
Boy/Male
Muslim
Competent
Girl/Female
Indian
Competent
Boy/Male
Anglo Saxon
Competent.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Female
Egyptian
, an uncertain goddess.
Boy/Male
Indian, Sanskrit
Competent
Surname or Lastname
English
English : variant spelling of See.
Boy/Male
Arabic, Muslim
Competent
Female
English
Short form of English Elizabeth, BET means "God is my oath."Â
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
Surname or Lastname
Portuguese
Portuguese : patronymic from the personal name Pedro (see Peter).Spanish and Jewish (Sephardic) : variant of Perez 2.English : variant of Pierce.Possibly also Hungarian : occupational name from peres ‘procurator’, ‘advocate’ (from per ‘trial’).
Girl/Female
Indian
Strength
Girl/Female
Hindu, Indian
In a Sastra
Girl/Female
Muslim/Islamic
Star
Girl/Female
Tamil
Adhvika | அதà¯à®µà¯€à®•à®¾
World, Earth, Unique
Boy/Male
Tamil
Nageshwar | நாகேஷà¯à®µà®°
Lord Shiva
Surname or Lastname
English
English : variant of Crass.
Girl/Female
Spanish American
Surname or Lastname
English
English : habitational name from any of various places called Carleton or Carlton, from Old Norse karl ‘common man’, ‘peasant’ + Old English tūn ‘settlement’ (compare Charlton 1). Places spelled Carl(e)ton (as opposed to Charlton) are in areas of Scandinavian settlement, mostly in northern England.Irish : Americanized and altered form of Carlin 1.
Boy/Male
English
ModernJaron 'cry of rejoicing.
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
v. t.
To provide with an implement or implements; to cause to be fulfilled, satisfied, or carried out, by means of an implement or implements.
v. t.
The interval wanting to complete the octave; -- the fourth is the complement of the fifth, the sixth of the third.
imp. & p. p.
of Set
v. t.
To cause to sit; to make to assume a specified position or attitude; to give site or place to; to place; to put; to fix; as, to set a house on a stone foundation; to set a book on a shelf; to set a dish on a table; to set a chest or trunk on its bottom or on end.
v. t.
A compliment.
n.
See Set, n., 2 (e) and 3.
v. t.
To compliment.
v. t.
To supply a lack; to supplement.
a.
Self-satisfied; contented; kindly; as, a complacent temper; a complacent smile.
v. t.
Full quantity, number, or amount; a complete set; completeness.
v. i.
To fit or suit one; to sit; as, the coat sets well.
n.
A number of things of the same kind, ordinarily used or classed together; a collection of articles which naturally complement each other, and usually go together; an assortment; a suit; as, a set of chairs, of china, of surgical or mathematical instruments, of books, etc.
v. i.
To pass compliments; to use conventional expressions of respect.
n.
That which is set, placed, or fixed.
a.
Fixed in position; immovable; rigid; as, a set line; a set countenance.
a.
Established; prescribed; as, set forms of prayer.
a.
Regular; uniform; formal; as, a set discourse; a set battle.
v. t.
To compose; to arrange in words, lines, etc.; as, to set type; to set a page.
v. t.
To praise, flatter, or gratify, by expressions of approbation, respect, or congratulation; to make or pay a compliment to.
n.
An expression, by word or act, of approbation, regard, confidence, civility, or admiration; a flattering speech or attention; a ceremonious greeting; as, to send one's compliments to a friend.