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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
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
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)
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
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
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
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
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
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)
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
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
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)
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)
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
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
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
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
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
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)
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
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
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)
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
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
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
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)
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)
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
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
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
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)
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
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
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
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
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
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
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)
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
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 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
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
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
On bipartite matching and vertex cover
the complement of a graph G is an independent set in G, and as we have already described an independent set in a bipartite graph G is a complement of a
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
Concept in axiomatic set theory
relative complement in positive set theory. In von Neumann–Bernays–Gödel set theory, a distinction is made between sets and classes. A class C is a set if and
Axiom_schema_of_specification
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
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible
Moschovakis_coding_lemma
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
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
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
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
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 mathematical logic
model theory, the basic notion of which is an abstract logic; the more general notion of an institution was later introduced, which advances from a set-theoretical
Lindström's_theorem
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
Technique invented by Paul Cohen for proving consistency and 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)
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
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
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
Binary operation combining the vertex and edge sets of two graphs
single-vertex graphs by a combination of disjoint union and complement operations. Join (graph theory) Graph minor Rosen, Kenneth H. (1999), Handbook of Discrete
Disjoint_union_of_graphs
Mathematical set that can be enumerated
be sets which are incomparable to N {\displaystyle \mathbb {N} } , the so-called Dedekind finite infinite sets. In 1874, in his first set theory article
Countable_set
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
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
Basic notion of sameness in mathematics
century, set theory (specifically Zermelo–Fraenkel set theory) became the most common foundation of mathematics. In set theory, any two sets are defined
Equality_(mathematics)
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
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
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
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
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)
Subset of a preorder that contains all larger elements
sets is again an upper set. The complement of an upper set is a lower set, and vice versa. Given a partially ordered set ( X , ≤ ) , {\displaystyle (X,\leq
Upper_and_lower_sets
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
In set theory, an extender is a system of ultrafilters which represents an elementary embedding witnessing large cardinal properties. A nonprincipal ultrafilter
Extender_(set_theory)
Infinite set that is not countable
these characterizations can be proved equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth
Uncountable_set
"Small" subset of a topological space
{\displaystyle X=[0,2]} the set [ 0 , 1 ] {\displaystyle [0,1]} is nonmeagre. But it is not comeagre, as its complement ( 1 , 2 ] {\displaystyle (1,2]}
Meagre_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
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
Yes-or-no question that cannot ever be solved by a computer
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct
Undecidable_problem
shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical
Mathematical_object
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
Surname or Lastname
English
English : variant spelling of See.
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Boy/Male
Muslim
Compliments, Happiness
Boy/Male
Indian, Sanskrit
Competent
Boy/Male
Arabic, Muslim
Competent
Female
Egyptian
, an uncertain goddess.
Girl/Female
Indian
Competent
Boy/Male
Muslim
Competent
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.
Boy/Male
Arabic, Muslim
Competent
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
Girl/Female
Indian
Competent.
Female
English
Short form of English Elizabeth, BET means "God is my oath."Â
Boy/Male
Japanese
Complacent; satisfied.
Male
English
Short form of English Stephen, STE means "crown."
Boy/Male
Anglo Saxon
Competent.
Boy/Male
Arabic, Muslim
Competent
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Boy/Male
Arabic, Muslim
Competent
Boy/Male
Hindi
Competent.
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
Male
Irish
Irish Gaelic form of French Édouard, EADBHÃRD means "guardian of prosperity."
Girl/Female
English
From the Old English Ealdraed, meaning old counsel. Aldred was common before the Norman Conquest,...
Female
Hebrew
(ש×ִפְרָה) Hebrew name SHIPHRAH means "beauty, brightness." In the bible, this is the name of two midwives.Â
Boy/Male
Indian
First Ray of Sun
Male
Danish
, brave warrior, or, hero.
Girl/Female
Australian, Chinese, Danish, Finnish, Swedish
Pious; Godly; Honorable
Girl/Female
Hindu, Indian, Malayalam, Marathi
One who Attracts the World; Goddess Durga
Girl/Female
Greek
Welcoming; hospitable.
Girl/Female
Hindu, Indian
Moon
Boy/Male
Arabic, Australian
Eighth Month of the Islamic Calender
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
COMPLEMENT SET-THEORY
v. t.
The interval wanting to complete the octave; -- the fourth is the complement of the fifth, the sixth of the third.
a.
Self-satisfied; contented; kindly; as, a complacent temper; a complacent smile.
v. t.
To compose; to arrange in words, lines, etc.; as, to set type; to set a page.
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. t.
To praise, flatter, or gratify, by expressions of approbation, respect, or congratulation; to make or pay a compliment to.
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.
To compliment.
v. i.
To pass compliments; to use conventional expressions of respect.
v. t.
Full quantity, number, or amount; a complete set; completeness.
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.
n.
See Set, n., 2 (e) and 3.
v. t.
To supply a lack; to supplement.
imp. & p. p.
of Set
v. i.
To fit or suit one; to sit; as, the coat sets well.
a.
Fixed in position; immovable; rigid; as, a set line; a set countenance.
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.
a.
Regular; uniform; formal; as, a set discourse; a set battle.
n.
That which is set, placed, or fixed.
v. t.
A compliment.
a.
Established; prescribed; as, set forms of prayer.