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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)
Branch of algebraic geometry
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties
Intersection_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)
Common elements of two or more sets
parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements
Intersection
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)
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
Shape formed from points common to other shapes
status of an operation with sets, intersection (set theory), in works by Giuseppe Peano. For the determination of the intersection point of two non-parallel
Intersection_(geometry)
Elements in exactly one of two sets
two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For
Symmetric_difference
Topics referred to by the same term
Intersection theory may refer to: Intersection theory, especially in algebraic geometry Intersection (set theory) This disambiguation page lists articles
Intersection theory (disambiguation)
Intersection_theory_(disambiguation)
Theory of discrimination
application of intersectionality. Patricia Hill Collins, author of Intersectionality as Critical Social Theory (2019), refers to the various intersections of social
Intersectionality
Topics referred to by the same term
Francisco Intersection in mathematics, including: Intersection (set theory), the set of elements common to some collection of sets Intersection (geometry)
Intersection_(disambiguation)
Identities and relationships involving sets
the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation
Algebra_of_sets
Secure multiparty computation cryptographic technique
Private set intersection is a secure multiparty computation cryptographic technique that allows two parties holding sets to compare encrypted versions
Private_set_intersection
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
determinacy Empty set Forcing (mathematics) Fuzzy set Hereditary set Internal set theory Intersection (set theory) Inner model theory Core model Covering
List_of_set_theory_topics
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
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
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
Graph representing intersections between given sets
In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets. Any graph can be represented as an
Intersection_graph
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)
instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of H j ( X ) {\displaystyle H_{j}(X)} is represented
Intersection_homology
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
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
Set whose elements all belong to another set
on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and
Subset
Family of subsets representing "large" sets
mathematics, a filter on a set is a collection of nonempty subsets which is closed under taking supersets and finite intersections. An example of filter is
Filter_on_a_set
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)
In mathematics, operation on sets
graphs Intersection (set theory) – Set of elements common to all of some sets List of set identities and relations – Equalities for combinations of sets Partition
Disjoint_union
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
Class of alternative set theories
In mathematical logic, positive set theory is the name for a class of alternative set theories in which the axiom of comprehension holds for at least the
Positive_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
Partial order with well-ordered predecessors
In set theory, a tree is a partially ordered set ( T , < ) {\displaystyle (T,<)} such that for each t ∈ T {\displaystyle t\in T} , the set { s ∈ T : s
Tree_(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
Area of discrete mathematics
geometric graph theory studies planar graphs, relationship to higher-dimensional convex polytopes, intersection of geometrical shaped sets, and other geometries'
Graph_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)
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
In mathematical set theory, a pseudo-intersection of a family of sets is an infinite set S such that each element of the family contains all but a finite
Pseudo-intersection
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
Sets with no element in common
set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are
Disjoint_sets
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
Shared independent set of two matroids
matroid intersection problem is to find a common independent set with the maximum possible weight. These problems generalize many problems in graph theory and
Matroid_intersection
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
Diagonal intersection is a term used in mathematics, especially in set theory. If δ {\displaystyle \displaystyle \delta } is an ordinal number and ⟨ X
Diagonal_intersection
determinacy Empty set Forcing (mathematics) Fuzzy set Internal set theory Intersection (set theory) L L(R) Large cardinal property Musical set theory Ordinal number
List of mathematical logic topics
List_of_mathematical_logic_topics
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
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
Algebraic concept in measure theory, also referred to as an algebra of sets
empty set as an element, and is closed under the operations of taking complements in X , {\displaystyle X,} finite unions, and finite intersections. Fields
Field_of_sets
Supposition or system of ideas intended to explain something
A theory is, in general, a set of propositions or ideas about something, developed in a variety of ways through any sort of reasoning. This includes informal
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
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)
Equivalence relation expressing that two elements have the same image under a function
iff any family of closed sets having fip has non-empty intersection at PlanetMath. Awodey, Steve (2010) [2006]. Category Theory. Oxford Logic Guides. Vol
Kernel_(set_theory)
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
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
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
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
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
A set intersection oracle (SIO) is a data structure which represents a collection of sets and can quickly answer queries about whether the set intersection
Set_intersection_oracle
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
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
Subfield of mathematics
Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic
Mathematical_logic
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)
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
Overview of and topical guide to logic
set Intension Intersection (set theory) Inverse function Large cardinal Löwenheim–Skolem theorem Map (mathematics) Multiset Morse–Kelley set theory Naïve
Outline_of_logic
Concept in set theory
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology, called the product topology. This space
Baire_space_(set_theory)
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
Two sets with a small overlap
In mathematics, two sets are almost disjoint if their intersection is small in some sense; different definitions of "small" will result in different definitions
Almost_disjoint_sets
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)
Fewest cliques covering a graph's edges
In the mathematical field of graph theory, the intersection number of a graph G = ( V , E ) {\displaystyle G=(V,E)} is the smallest number of elements
Intersection number (graph theory)
Intersection_number_(graph_theory)
Property in general topology
{\displaystyle {\mathcal {A}}} of subsets of a set X {\displaystyle X} is said to have the finite intersection property (FIP) if any finite subfamily of A
Finite_intersection_property
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
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
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
Mathematical ways to group elements of a set
The sets in P are said to exhaust or cover X. See also collectively exhaustive events and cover (topology). The intersection of any two distinct sets in
Partition_of_a_set
Pair of logical equivalences
change the operator when doing a substitution. In set theory, it is often stated as "union and intersection interchange under complementation", which can
De_Morgan's_laws
Notion in computational learning
of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays
Shattered_set
Axioms for the natural numbers
defined as the intersection of all sets closed under s that contain the empty set. Each natural number is equal (as a set) to the set of natural numbers
Peano_axioms
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
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
Simple theorems in the algebra of sets
Simple_theorems_in_the_algebra_of_sets
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
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
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
In geometry, set whose intersection with every line is a single line segment
convex. The boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset A of Euclidean
Convex_set
1960 mathematics textbook by Paul Halmos
Naive Set Theory is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory. Originally published by Van Nostrand
Naive_Set_Theory_(book)
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
Family closed under unions and relative complements
called a ring (of sets) if it is closed under union and intersection. That is, the following two statements are true for all sets A {\displaystyle A}
Ring_of_sets
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
Topics referred to by the same term
Unicode); also ∩ {\displaystyle \cap } , the mathematical symbol for Intersection (set theory) A shape used to describe narrative structure, specifically the
Inverted_U
Topics referred to by the same term
may refer to: In set theory, an overlap of elements shared between sets is called an intersection, as in a Venn diagram. In music theory, overlap is a synonym
Overlap
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
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
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
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)
Branch of mathematics
upon the concepts of set theory, arithmetic, and binary relations. Orders are special binary relations. Suppose that P is a set and that ≤ is a relation
Order_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
Generalized notion of counting curve intersections
a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory. Let X be a
Intersection_number
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
Concerned with the notion of stability in model theory
strongly minimal sets, such as intersection theory, Zilber proved a weak form of the Trichotomy Conjecture for uncountably categorical theories. Although Ehud
Stable_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
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)
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
System of mathematical set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard
Internal_set_theory
INTERSECTION SET-THEORY
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INTERSECTION SET-THEORY