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Use of braces for specifying sets
set of all even integers, expressed in set-builder notation. In mathematics and more specifically in set theory, set-builder notation is a notation for
Set-builder_notation
Collection of mathematical objects
{\displaystyle \{0,1,-1,2,-2,3,-3,\ldots \}.} Set-builder notation specifies a set as being the set of all elements that satisfy some logical formula
Set_(mathematics)
Mathematical set formed from two given sets
where a is an element of A and b is an element of B. In terms of set-builder notation, that is A × B = { ( a , b ) ∣ a ∈ A and b ∈ B } . {\displaystyle
Cartesian_product
Elements in exactly one of two sets
using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation: A Δ B = { x : ( x ∈ A ) ⊕ ( x ∈ B ) } . {\displaystyle A\mathbin
Symmetric_difference
Syntactic construct for creating a list based on existing lists
mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions. Consider the following example in mathematical set-builder
List_comprehension
Informal set theories
has blonde hair} denotes the set of everything with blonde hair. This notation is called set-builder notation (or "set comprehension", particularly in
Naive_set_theory
Set of elements in any of some sets
the set of elements which are in A, in B, or in both A and B. In set-builder notation, A ∪ B = { x : x ∈ A or x ∈ B } {\displaystyle A\cup B=\{x:x\in
Union_(set_theory)
2. Set-builder notation for a singleton set: { x } {\displaystyle \{x\}} denotes the set that has x as a single element. {□, ..., □} Set-builder notation:
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Set of elements common to all of some sets
of the collection M {\displaystyle M} is defined as the set (see set-builder notation) ⋂ A ∈ M A = { x : for all A ∈ M , x ∈ A } . {\displaystyle \bigcap
Intersection_(set_theory)
Paradox in set theory
a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory
Russell's_paradox
Association of one output to each input
concept of a relation, but using more notation (including set-builder notation): A function is formed by three sets (often as an ordered triple), the domain
Function_(mathematics)
Mathematical set containing no elements
the empty set, but this is now considered to be an improper use of notation. The symbol ∅ is available at Unicode point U+2205 ∅ EMPTY SET. It can be
Empty_set
Any one of the distinct objects that make up a set in set theory
expressed notationally as 3 ∈ A {\displaystyle 3\in A} . Writing A = { 1 , 2 , 3 , 4 } {\displaystyle A=\{1,2,3,4\}} means that the elements of the set A are
Element_of_a_set
Branch of mathematics that studies sets
mathematics. Set theory begins with a fundamental binary relation between an object o and a set A. If o is an element (or member) of A, the notation o ∈ A is
Set_theory
Set of the elements not in a given subset
this notation can be ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all
Complement_(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
Collection of sets in mathematics that can be defined based on a property of its members
satisfies Φ {\displaystyle \Phi } may be expressed with the shorthand notation Φ ( x ) = y {\displaystyle \Phi (x)=y} . Another approach is taken by the
Class_(set_theory)
Set with exactly one element
a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set { 0 } {\displaystyle \{0\}} is a singleton
Singleton_(mathematics)
Set of elements that commute with every element of a group
is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation, Z(G) =
Center_(group_theory)
All numbers between two given numbers
notations are described in International standard ISO 31-11. Thus, in set builder notation, ( a , b ) = ] a , b [ = { x ∈ R ∣ a < x < b } , [ a , b ) = [ a
Interval_(mathematics)
Set whose elements all belong to another set
{\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
Mathematical set of all subsets of a set
the set S is n), then the number of all the subsets of S is |P(S)| = 2n. This fact, as well as the reason for the notation 2S denoting the power set P(S)
Power_set
Convention where symbols represent concepts
Z notation, a formal notation for specifying objects using Zermelo–Fraenkel set theory and first-order predicate logic Ordinal notation Set-builder notation
Notation_system
Set theory concept
{\displaystyle n} , the set Vn+1 contains 2 ↑↑ n {\displaystyle 2\uparrow \uparrow n} elements using Knuth's up-arrow notation. So the finite stages of
Von_Neumann_universe
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
Typographic symbol
{\displaystyle A} , to just A {\displaystyle A} set-builder notation: { x | x < 2 } {\displaystyle \{x|x<2\}} , read "the set of x such that x is less than two".
Vertical_bar
Concept in axiomatic set theory
} . By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as B = { x ∈ A ∣ φ ( x ) } {\displaystyle
Axiom_schema_of_specification
Mathematical concept
elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle
Equivalence_class
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
Universal_set
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
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
Standard system of axiomatic set theory
ordinal rank.[citation needed] Subsets are commonly constructed using set builder notation. For example, the even integers can be constructed as the subset
Zermelo–Fraenkel_set_theory
Size of a set in mathematics
\beth } , the second letter of the Hebrew alphabet) provide a concise notation for powersets of the real numbers starting from ℶ 0 = | N | {\displaystyle
Cardinality
Pair of logical equivalences
I is some, possibly countably or uncountably infinite, indexing set. In set notation, De Morgan's laws can be remembered using the mnemonic "break the
De_Morgan's_laws
Topics referred to by the same term
abstraction and the law of abstraction in formal logic Set abstraction (AKA set comprehension, set-builder notation) Hardware abstraction, an abstraction layer on
Abstraction_(disambiguation)
Reversal of the order of elements of a binary relation
yL^{\operatorname {T} }x} if and only if x L y . {\displaystyle xLy.} In set-builder notation, L T = { ( y , x ) ∈ Y × X : ( x , y ) ∈ L } . {\displaystyle L^{\operatorname
Converse_relation
based on existing lists. It follows the form of the mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions
Comparison of programming languages (list comprehension)
Comparison_of_programming_languages_(list_comprehension)
Set of points that satisfy some specified conditions
equation 2x + 3y – 6 = 0. Algebraic variety Curve Line (geometry) Set-builder notation Shape (geometry) James, Robert Clarke; James, Glenn (1992), Mathematics
Locus_(mathematics)
Property or quality connoted by a word, phrase, or another symbol
Montague grammar Ostension Temperature paradox Sense and reference Set-builder notation Antony Flew (1979). Dictionary of Philosophy. p. 117. Putnam, Hilary
Intension
Set-builder notation Set-theoretic topology Simple theorems in the algebra of sets Subset Θ (set theory) Tree (descriptive set theory) Tree (set theory)
List_of_set_theory_topics
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
Set of tuples in mathematical logic that satisfy a predicate
"Tuesday follows the weekday Saturday" (for example) is false. Using set-builder notation, the extension of the n-ary predicate Φ {\displaystyle \Phi } can
Extension_(predicate_logic)
Punctuation mark with two dots (:)
definitions. In mathematical logic, when using set-builder notation for describing the characterizing property of a set, it is used as an alternative to a vertical
Colon_(punctuation)
Identities and relationships involving sets
mathematics, particularly in the study of set theory, the algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection
Algebra_of_sets
Generalization of "n-th" to infinite cases
unbounded set in κ {\displaystyle \kappa } . Intuitively, stationary sets are "large" enough that they cannot be avoided by any club set. Using the notation of
Ordinal_number
Czech mathematician
ˈjɛx]; born 29 January 1944 in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years. He was educated at Charles
Thomas_Jech
Family of subsets representing "large" sets
In 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
Filter_on_a_set
Set of the values of a function
) {\displaystyle f(A)} when there is no risk of confusion. Using set-builder notation, this definition can be written as f [ A ] = { f ( a ) : a ∈ A }
Image_(mathematics)
Concept in set theory
F_{P}} , and denoted F P [ A ] {\displaystyle F_{P}[A]} or (using set-builder notation) { F P ( x ) : x ∈ A } {\displaystyle \{F_{P}(x):x\in A\}} . The
Axiom_schema_of_replacement
Any collection of sets, or subsets of a set
"family of sets" because if one instead uses "set of sets" then the subsequent use of "set" can be confusing as to whether it is the containing set or one
Family_of_sets
In mathematics, operation on sets
bijection. In this context, the notation ∐ i ∈ I A i {\textstyle \coprod _{i\in I}A_{i}} is often used. The disjoint union of two sets A {\displaystyle A} and
Disjoint_union
Infinite set not splittable into infinite sets
In set theory, an amorphous set is an infinite set that is not the disjoint union of two infinite subsets. Amorphous sets cannot exist if the axiom of
Amorphous_set
Finite ordered list of elements
n-tuple can be formally defined as the image of a function that has the set of the first n natural numbers as its domain (1, 2, ..., n). Tuples may be
Tuple
Sets whose elements have degrees of membership
In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an
Fuzzy_set
Computers accessing information referentially
could be β, γ, δ, ... or η→π, ς ∨ σ, ... When set-builder notation is employed the statement Δ={α} means the set of all formulae — so although the reference
Indirection
System of mathematical set theory
steps of collection of sets, followed by a restriction through separation. All results are also expressed using set builder notation. Firstly, given b {\displaystyle
Kripke–Platek_set_theory
A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian
Nested_set_collection
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
Concept in axiomatic set theory
power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a set P ( x
Axiom_of_power_set
Concept that is not defined in terms of previously defined concepts
functions as primitive, as well as the phrase "such that" as used in set builder notation. (pp 18,9) Regarding relations, Russell takes as primitive notions
Primitive_notion
System of mathematical set theory
this notational convention is only mnemonic. The monadic predicate M x , {\displaystyle Mx,} whose intended reading is "the class x is a set", abbreviates
Morse–Kelley_set_theory
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
Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere
List_of_types_of_sets
Mathematician (1845–1918)
and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter ℵ {\displaystyle \aleph
Georg_Cantor
Alternative to the standard Zermelo–Fraenkel set theory
set theory Morse–Kelley set theory Tarski–Grothendieck set theory Ackermann set theory Type theory New Foundations Positive set theory Internal set theory
List of alternative set theories
List_of_alternative_set_theories
Origin and evolution of the symbols used to write equations and formulas
symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators
History of mathematical notation
History_of_mathematical_notation
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
3-volume treatise on mathematics, 1910–1913
logic using the most convenient notation that precise expression allows; to solve the paradoxes that plagued logic and set theory at the turn of the 20th
Principia_Mathematica
System of mathematical set theory
Zermelo 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
Zermelo_set_theory
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
Abraham_Fraenkel
Finite collection of distinct objects
2-subset of it. This notation { 1 , ⋯ , n } {\displaystyle \{1,\cdots ,n\}} may be defined recursively as { 1 , ⋯ , n } = { ∅ (the empty set) if n = 0 { 1
Finite_set
Mathematical use of "for all" and "there exists"
that n2 ≤ 4 are in {0,1,2}." The same construct is expressible in set-builder notation as { n ∈ N : n 2 ≤ 4 } = { 0 , 1 , 2 } . {\displaystyle \{n\in \mathbb
Quantifier_(logic)
Mathematician and philosopher (1906–1978)
Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier
Kurt_Gödel
In mathematics, invertible homomorphism
one (in set builder notation), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets { 4 , 5
Isomorphism
System of mathematical set theory
extensionality implies the uniqueness of the set p {\displaystyle p} , which allows us to introduce the notation { x , y } . {\displaystyle \{x,y\}.} Ordered
Von Neumann–Bernays–Gödel set theory
Von_Neumann–Bernays–Gödel_set_theory
Axiom used in set theory
Extensional context Extension (predicate logic) Set theory Glossary of set theory In the original notation, ( α = β ) {\displaystyle (\alpha =\beta )} for
Axiom_of_extensionality
Finite sets whose elements are all hereditarily finite sets
up-arrow notation (a tower of n − 1 {\displaystyle n-1} powers of two), and the union of countably many finite sets is countable. Equivalently, a set is hereditarily
Hereditarily_finite_set
Form of type polymorphism
can be expressed using Set-builder notation, which uses a predicate to define a set. Predicates can be defined over a domain (set of possible values) D
Subtyping
System of mathematical set theory
and then adds “Tarski's axiom”. We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully
Tarski–Grothendieck set theory
Tarski–Grothendieck_set_theory
German mathematician (1831–1916)
Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as logicism. Dedekind's
Richard_Dedekind
Axiom of set theory
axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order
Axiom_of_regularity
American mathematician (1934–2007)
hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. Cohen was born in Long Branch
Paul_Cohen
Vectors mapped to 0 by a linear map
The dimension of the kernel of A is called the nullity of A. In set-builder notation, N ( A ) = Null ( A ) = ker ( A ) = { x ∈ K n ∣ A x = 0 }
Kernel_(linear_algebra)
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 concept for comparing objects
class. Various notations are used in the literature to denote that two elements a {\displaystyle a} and b {\displaystyle b} of a set are equivalent with
Equivalence_relation
One-to-one correspondence
function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given
Bijection
Symbol representing a property or relation in logic
functions (i.e., functions from a set element to a truth value). Set-builder notation makes use of predicates to define sets. In autoepistemic logic, which
Predicate_(logic)
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
Axiom_of_infinity
Size of a possibly infinite set
measures the cardinality of a set, i.e., how many elements there are in a set. The cardinal number associated with a set A {\displaystyle A} is generally
Cardinal_number
Branch of mathematics
empty set is the least set under the subset order. Formally, an element m is a least element if: m ≤ a, for all elements a of the order. The notation 0 is
Order_theory
Structure of a formal language
symbols. For these examples, formal languages are specified using set-builder notation. Consider the grammar G {\displaystyle G} where N = { S , B } {\displaystyle
Formal_grammar
Mathematical concept
this principle is also true for arbitrary well-ordered sets, but since any well-ordered set can be indexed by ordinals in an order-preserving way, it
Transfinite_induction
Proof in set theory
infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some
Cantor's_diagonal_argument
Axiomatic set theories based on the principles of mathematical constructivism
(}y=x\leftrightarrow Q(y){\big )}} . As is also common, one makes use set builder notation for classes, which, in most contexts, are not part of the object
Constructive_set_theory
Template that specifies one or more axioms
image of any set under that definable function is also a set. Richard Montague proved a non-finite axiomatizability result for ZF-style set theories; in
Axiom_schema
Operation in algebra and mathematics
such that their sum is equal to 1 {\displaystyle 1} . In set-builder notation, this is the set D ( X ) = { f : X → [ 0 , 1 ] : # supp ( f ) < + ∞ ∑ x ∈
Monad_(category_theory)
Proof by Alan Turing
his development of typewriter-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine"
Turing's_proof
Finding values for variables that make an equation true
is not just one solution, but an infinite set of solutions, which can be written using set builder notation as { ( x , y , z ) ∣ 3 x + 2 y − 21 z = 0
Equation_solving
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
German logician and mathematician (1871–1953)
mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929
Ernst_Zermelo
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SET BUILDER-NOTATION
SET BUILDER-NOTATION
Male
Hindi/Indian
(सेठ) Hindi name derived from the Sanskrit word setu, SETH means "bridge." Compare with other forms of Seth.
Male
Scandinavian
Scandinavian form of Old Norse Baldr, BALDER means "lord, prince." In mythology, this is the name of a son of Odin and Frigg.
Boy/Male
British, English
Wagon-builder
Surname or Lastname
English, German, Danish, and Jewish (Ashkenazic)
English, German, Danish, and Jewish (Ashkenazic) : variant of Wild.Thomas Wilder is recorded as a freeman of Charlestown, MA, in 1640. He had numerous prominent descendents.
Surname or Lastname
English
English : metonymic occupational name for a spoon maker, from Old French cuiller ‘spoon’, ‘ladle’.
Surname or Lastname
English (mainly Sussex and Kent)
English (mainly Sussex and Kent) : topographic name from Middle English hilder ‘dweller on a slope’ (from Old English hylde ‘slope’).
Boy/Male
Arabic, British, English
Builder; Architect
Boy/Male
Arabic Muslim
Long-living; builder.
Boy/Male
British, English
Wagon-builder
Female
Egyptian
, an uncertain goddess.
Boy/Male
British, English
Wagon Builder
Boy/Male
British, English
Wagon-builder
Male
English
Short form of English Stephen, STE means "crown."
Biblical
whom Jehovah will build up;God builds;Jehovah is builder;
Boy/Male
British, English
Wagon-builder
Boy/Male
Egyptian Hebrew Swedish
Son of Seb and Nut.
Surname or Lastname
English
English : variant spelling of See.
Surname or Lastname
South German
South German : probably an occupational name for a gauger or sealer of barrels, from an agent derivative of Middle High German beil ‘barrel inspection’. See also Beiler.Altered spelling of Böhler (see Boehler).English : variant spelling of Bailor.
Boy/Male
American, German, Hebrew
Strength; Builder; Eternal
Male
Hebrew
Variant spelling of Hebrew Sheth, SHET means "buttocks."
SET BUILDER-NOTATION
SET BUILDER-NOTATION
SET BUILDER-NOTATION
SET BUILDER-NOTATION
SET BUILDER-NOTATION
SET BUILDER-NOTATION
SET BUILDER-NOTATION
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