Category: Basic concepts in set theory

Set function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are the total functions from A
Inverse function
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exist
Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For examp
K-cell (mathematics)
A -cell is a higher-dimensional version of a rectangle or rectangular solid. It is the Cartesian product of closed intervals on the real line. This means that a -dimensional rectangular solid has each
Index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled by means of the elements of a set J, the
Indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset
Kernel (set theory)
In set theory, the kernel of a function (or equivalence kernel) may be taken to be either * the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: X → Y. The term rang
Intersection (set theory)
In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to
Predicate (mathematical logic)
In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula , the symbol is a predicate which applies to the individual constant . Similarly,
Disjoint sets
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5,
Map (mathematics)
In mathematics, a map is often used as a synonym for a function, but may also refer to some generalizations. Originally, this was an abbreviation of mapping, which often refers to the action of applyi
Unordered pair
In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an order
Bijection, injection and surjection
In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from
Domain of a function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by or , where f is the function. More precisely, given a function , the domain of f is X
Complement (set theory)
In set theory, the complement of a set A, often denoted by A∁ (or A′), is the set of elements not in A. When all sets in the universe, i.e. all sets under consideration, are considered to be members o
Identity function
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. T
Indexed family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of inte
Inclusion map
In mathematics, if is a subset of then the inclusion map (also inclusion function, insertion, or canonical injection) is the function that sends each element of to treated as an element of A "hooked a
Pointed set
In mathematics, a pointed set (also based set or rooted set) is an ordered pair where is a set and is an element of called the base point, also spelled basepoint. Maps between pointed sets and – calle
Empty set
In mathematics, the empty 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 that the empty set exists by includ
Horizontal line test
In mathematics, the horizontal line test is a test used to determine whether a function is injective (i.e., one-to-one).
Range of a function
In mathematics, the range of a function may refer to either of two closely related concepts: * The codomain of the function * The image of the function Given two sets X and Y, a binary relation f be
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, f(x1) = f(x2) implies x
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. Howeve
Large set (Ramsey theory)
In Ramsey theory, a set S of natural numbers is considered to be a large set if and only if Van der Waerden's theorem can be generalized to assert the existence of arithmetic progressions with common
Disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets is a set often denoted by with an injection of each into such that the images of these injections form a partition of (tha
History of the function concept
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-
List of set identities and relations
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It
Element (mathematics)
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for eac
Projection (set theory)
In set theory, a projection is one of two closely related types of functions or operations, namely: * A set-theoretic operation typified by the jth projection map, written , that takes an element of
Function (mathematics)
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the fun
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defin
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function) is a function f that every element y can be mapped from element x so that f(x) = y. In other words, every element of
Indicator vector
In mathematics, the indicator vector or characteristic vector or incidence vector of a subset T of a set S is the vector such that if and if If S is countable and its elements are numbered so that , t
Algebra of sets
In mathematics, the algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersect
Symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symme
List of types of sets
Sets can be classified according to the properties they have.
Union (set theory)
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 through which sets can be combined and related to
Singleton (mathematics)
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set is a singleton whose single element is .
In mathematics, set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper sub
Jouanolou's trick
In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine fibers from an affine variety W to X. The variety W is therefo
Inhabited set
In constructive mathematics, a set is inhabited if there exists an element In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in intuitionisti
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is pai
Saturated set
In mathematics, particularly in the subfields of set theory and topology, a set is said to be saturated with respect to a function if is a subset of 's domain and if whenever sends two points and to t
Fiber (mathematics)
In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context: 1. * In naive set theory, the fiber of the element in the set under a map is th
In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referre
Ordered pair
In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless
Function composition
In mathematics, function composition is an operation  ∘  that takes two functions f and g, and produces a function h = g  ∘  f such that h(x) = g(f(x)). In this operation, the function g is applied to
Smooth maximum
In mathematics, a smooth maximum of an indexed family x1, ..., xn of numbers is a smooth approximation to the maximum function meaning a parametric family of functions such that for every α, the funct
Choice function
A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns some element of each set S in that collection to S by f(S); f(S)
Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function at each element of a given subset of its domain produces a set, call