Searches , social queries for COUNTABLE SET
Search references for COUNTABLE SET. Phrases containing COUNTABLE SET
See searches and references containing COUNTABLE SET!
Search references for COUNTABLE SET. Phrases containing COUNTABLE SET
See searches and references containing COUNTABLE SET!COUNTABLE SET
Mathematical set that can be enumerated
mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers. Equivalently, a set is countable
Countable_set
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. The inductive definition above is well-founded
Hereditarily_countable_set
Measurable set whose measure is zero
covered by a countable union of intervals of arbitrarily small total length. A null set is not to be confused with the empty set as defined in set theory.
Null_set
Concept in mathematics
countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets
Axiom_of_countable_choice
Class of mathematical sets
{\displaystyle X} 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
Borel_set
Property in descriptive set theory
mathematical field of descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset
Perfect_set_property
Countable union of closed sets
In general topology, an Fσ set (pronounced F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French:
Fσ_set
Generalization of mass, length, area and volume
{\displaystyle X} is countable; and semifinite (without regard to whether X {\displaystyle X} is countable). (Thus, counting measure, on the power set P ( X ) {\displaystyle
Measure_(mathematics)
Generalization of "n-th" to infinite cases
discrete sets, so they are countable. Proof of first theorem: If P(α) = ∅ for some index α, then P′ is the countable union of countable sets. Therefore
Ordinal_number
Mapping function
infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n
Sigma-additive_set_function
Collection of mathematical objects
either finite sets or countably infinite sets (sets of cardinality ℵ0); some authors use "countable" to mean "countably infinite". Sets with cardinality
Set_(mathematics)
Set with no meaningful size
provides the basis of Riemann integration. Countable additivity, which applies to a countably infinite collection of sets, is a stronger axiom that underlies
Non-measurable_set
Set with an equinumerous proper subset
ZF) conditions: it has a countably infinite subset; there exists an injective map from a countably infinite set (say, N, the set of all natural numbers)
Dedekind-infinite_set
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
Branch of mathematics that studies sets
Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included
Set_theory
On convergent subsequences of functions that are locally of bounded total variation
most countably many points of discontinuity. Let An={x∈I;fn(y)↛fn(x) as y→x} be the set of discontinuities of fn; each of these sets are countable by the
Helly's_selection_theorem
Broadest definition of sizes in integer-dimensional spaces
other kinds of countable limits of sets. For example, every countable subset of the real line has Lebesgue measure zero, being a countable union of points
Lebesgue_measure
Index of articles associated with the same name
mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties
Axiom_of_countability
Property of mathematical sets
cocountable subset of a set X is a subset Y whose complement in X is a countable set. In other words, Y contains all but countably many elements of X. Since
Cocountability
Topology made of cocountable subsets
known as the countable complement topology, is a topology that can be defined on any infinite set X {\displaystyle X} . In this topology, a set is open if
Cocountable_topology
Maximal proper filter
sets is a countable set. However, ZF with the ultrafilter lemma is too weak to prove that a countable union of countable sets is a countable set. The Hahn–Banach
Ultrafilter_on_a_set
Finite sets whose elements are all hereditarily finite sets
H_{\aleph _{0}}} to be a set can also prove it to be countable. In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers
Hereditarily_finite_set
Set of elements common to all of some sets
also be written "A1∩A2∩A3∩⋯". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras
Intersection_(set_theory)
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)
Branch of topology
infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When
General_topology
set of formulas in the Lévy hierarchy. ρ The rank of a set. σ countable, as in σ-compact, σ-complete and so on. Σ 1. A sum of cardinals. 2. A set of
Glossary_of_set_theory
Size of a set in mathematics
6,⋯} and the set of rational numbers are countable. Uncountable sets are those strictly larger than the set of natural numbers. The set of all real numbers
Cardinality
Type of cardinal number in mathematics
_{1}} are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So ℵ 1 {\displaystyle
Regular_cardinal
In mathematics, a non-algebraic number
numbers are countable, while the real numbers ℝ and complex numbers ℂ are both uncountable, and therefore larger than any countable set. All transcendental
Transcendental_number
Algebraic structure of set algebra
a set X {\displaystyle X} is a nonempty collection Σ {\displaystyle \Sigma } of subsets of X {\displaystyle X} closed under complement, countable unions
Σ-algebra
Mathematical concept
be extended to an arbitrary countable set A (e.g. the set of n-tuples of integers, the set of rational numbers, the set of formulas in some formal language
Arithmetical_set
Inequality applying to probability spaces
inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is
Boole's_inequality
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
Axiomatic set theories based on the principles of mathematical constructivism
The empty set is not inhabited but generally deemed countable too, and note that the successor set of any countable set is countable. The set ω {\displaystyle
Constructive_set_theory
Every countable set is a strong measure zero set, and so is every union of countably many strong measure zero sets. Every strong measure zero set has Lebesgue
Strong_measure_zero_set
Mathematical set containing no elements
the 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
Empty_set
complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities. Given a pointset A contained in some product
Scale (descriptive set theory)
Scale_(descriptive_set_theory)
"Small" subset of a topological space
topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets whose
Meagre_set
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
Borel sets on spaces without a countable base for the topology. In practice, the use of Baire measures on Baire sets can often be replaced by the use
Baire_set
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
{\displaystyle I} denotes its index set) has a convergent subsequence if and only if there exists a countable set K ⊆ I {\displaystyle K\subseteq I} such
Bolzano–Weierstrass_theorem
Description of continuous random distribution
of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables
Probability_density_function
Mathematical lemma
is a countable set of dense subsets of P then there exists a D-generic filter F in P such that p ∈ F. Let p ∈ P be given. Since D is countable, D = { Di |
Rasiowa–Sikorski_lemma
Axiom of set theory
numbers are countable: As pointed out above, to show that a countable union of countable sets is itself countable requires the Axiom of countable choice.
Axiom_of_choice
Set of all limit points of a set
states that any Polish space can be written as the union of a countable set and a perfect set. Because any Gδ subset of a Polish space is again a Polish
Derived_set_(mathematics)
Technique for proving independence results
within M {\displaystyle M} (e.g. the countability of M {\displaystyle M} ), and thus prove the existence of sets that are "too complex for M {\displaystyle
Forcing_(mathematics)
Ordered listing of items in collection
is sometimes used for countable sets. However it is also often used for computably enumerable sets, which are the countable sets for which an enumeration
Enumeration
Set with operations obeying given axioms
which may be considered an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all
Algebraic_structure
Mathematician (1845–1918)
Cantor 1874 A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this
Georg_Cantor
Standard system of axiomatic set theory
{\displaystyle {\mathcal {A}}} is a set consisting of: A countably infinite number of variables used for representing sets The logical connectives ¬ {\displaystyle
Zermelo–Fraenkel_set_theory
Any collection of sets, or subsets of a set
sets δ-ring – Ring closed under countable intersections Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets Generalized
Family_of_sets
related to set theory. Algebra of sets Axiom of choice Axiom of countable choice Axiom of dependent choice Zorn's lemma Axiom of power set Boolean-valued
List_of_set_theory_topics
Use of braces for specifying sets
{Z} ,n=2k\}} — The set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation
Set-builder_notation
Infinite set that is not countable
mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related
Uncountable_set
Study of discrete mathematical structures
characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However
Discrete_mathematics
Concept in mathematical logic
set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set. Hereditarily countable set
Hereditary_set
Process of mapping a continuous set to a countable set
of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite number of elements
Quantization (signal processing)
Quantization_(signal_processing)
Anticommutating number
finite number of generators, typically n = 1, 2, 3 or 4, and those with a countably-infinite number of generators. These two situations are not as unrelated
Grassmann_number
Identities and relationships involving sets
algebra of sets, completed to include countably infinite operations. Axiomatic set theory Image (mathematics) § Properties Field of sets List of set identities
Algebra_of_sets
Finite collection of distinct objects
finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite"
Finite_set
Smallest ordinal number that, considered as a set, is uncountable
the order type of an uncountable well-ordered set. It is the supremum (least upper bound) of all countable ordinals. In the von Neumann representation,
First_uncountable_ordinal
Minimal standard model of ZFC
Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named;
Minimal_model_(set_theory)
Set of points on a line segment with certain topological properties
set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product 2_ℕ of the discrete two-point space 2_. By a
Cantor_set
Mathematical function for the probability a given outcome occurs in an experiment
x}p(\omega ).} The points where the cdf jumps always form a countable set; this may be any countable set and thus may even be dense in the real numbers. A discrete
Probability_distribution
{\displaystyle X} is called countably generated if the topology of X {\displaystyle X} is determined by the countable sets in a similar way as the topology
Countably_generated_space
Sets can be classified according to the properties they have. Empty set Finite set, Infinite set Countable set, Uncountable set Power set Closed set Open
List_of_types_of_sets
First article on transfinite set theory
theorems is his "revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's
Cantor's first set theory article
Cantor's_first_set_theory_article
Particular class of sets which can be described entirely in terms of simpler sets
H_{\alpha }} is the set of sets which are hereditarily of cardinality less than α {\displaystyle \alpha } (see hereditarily countable set#Generalizations)
Constructible_universe
Certain topology in mathematics
sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable; this union is an upper bound of the elements
Order_topology
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)
are fit to discrete data—data that could potentially take on only a countable set of values, such as the integers, and which are not infinitely divisible
Discrete_modelling
Subset that is closed and has no isolated points
can be uniquely written as the disjoint union of a perfect set and a countable set. This is also true more generally for all closed subsets of Polish spaces
Perfect_set
Paradox in set theory
existence of countable models (Skolem's paradox), but it enjoys some important advantages." In ZFC, given a set A, it is possible to define a set B that consists
Russell's_paradox
Topological space whose topology has a countable base
topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly
Second-countable_space
Mathematical logic concept
is the apparent contradiction that a countable model of first-order set theory could contain an uncountable set. The paradox arises from part of the Löwenheim–Skolem
Skolem's_paradox
Countable intersection of open sets
set is a subset of a topological space that is a countable intersection of open sets. The notation originated from the German nouns Gebiet 'open set'
Gδ_set
Point of a subset S around which there are no other points of S
injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which
Isolated_point
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Function from sets to numbers
many sets and generalized series. In particular, this is why the definition of "countably additive" is rarely extended from countably many sets F 1 ,
Set_function
Concept in topology
perfect set and a countable set. Moreover, if X {\displaystyle X} is uncountable, it can be written as the disjoint union of a perfect set and a countable open
Polish_space
Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real
Discrete_measure
Monotone maps have countable discontinuities
(monotone) function are necessarily jump discontinuities and there are at most countably many of them. Usually, this theorem appears in literature without a name
Discontinuities of monotone functions
Discontinuities_of_monotone_functions
Size of a possibly infinite set
infinite sets (for example the set of natural numbers N = {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection with N denumerable (countably infinite)
Cardinal_number
Quotient of two integers
{2}}} ), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers
Rational_number
Problem of finding the best feasible solution
object such as an integer, permutation or graph must be found from a countable set. A problem with continuous variables is known as a continuous optimization
Optimization_problem
Set of real numbers that is not Lebesgue measurable
the set of rational numbers because a single point has measure zero, and the set of rational numbers comprises a countable infinity of points. Any set which
Vitali_set
Proposition in mathematical logic
integers, and therefore the set of rational numbers is the same size (cardinality) as the set of integers: they are both countable sets. Cantor gave two proofs
Continuum_hypothesis
Area of mathematical logic
quasiminimally excellent classes are those in which every definable set is either countable or co-countable. They are key to the model theory of the complex exponential
Model_theory
to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that F is necessarily itself countable. Paul
Wetzel's_problem
Set whose elements all belong to another set
In mathematics, a set A is a subset of a set B if and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A
Subset
Theorem in set-theoretic geometry
G-equidecomposable sets may be found whose "sizes" vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many
Banach–Tarski_paradox
Mathematical concept
(that is, any set X {\displaystyle X} along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X {\displaystyle
Counting_measure
Number used for counting
product of primes Countable set – Mathematical set that can be enumerated Sequence – Function of the natural numbers in another set Ordinal number – Generalization
Natural_number
Proof in set theory
proof by contradiction to show that: The set T is uncountable. The proof starts by assuming that T is countable. Then all its elements can be written in
Cantor's_diagonal_argument
Elements in exactly one of two sets
symmetric difference of 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
Symmetric_difference
where each formula is of degree 1 and contains no uses of equality. For a countable collection of formulas φ i {\displaystyle \varphi ^{i}} of degree 1, we
Original proof of Gödel's completeness theorem
Original_proof_of_Gödel's_completeness_theorem
Model of computation over real numbers
latter are by definition restricted to a finite set of symbols. A Turing machine can represent a countable set (such as the rational numbers) by strings of
Blum–Shub–Smale_machine
Infinite cardinal number
(this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ 0
Aleph_number
Mathematical logic concept
there are countable transitive models of ZFC' (this is provable in ZFC), and the set of real numbers in such a model will be a countable set. The paradox
Absoluteness_(logic)
Any one of the distinct objects that make up a set in set theory
mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four
Element_of_a_set
travel, tourism, insurance
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
COUNTABLE SET
travel, tourism, insurance