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set builder notation wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 sets defined by a predicate toggle sets defined by a predicate subsection 1 1 specifying the domain 1 2 examples 2 more complex expressions on the left side of the notation 3 equivalent predicates yield equal sets 4 set existence axiom 5 in programming languages 6 see also 7 notes toggle the table of contents set builder notation 10 languages বাংলা español bahasa indonesia íslenska 日本語 한국어 русский simple english українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia use of braces for specifying sets n k z n 2 k displaystyle n mid exists k in mathbb 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 is a notation for specifying a set by a property that characterizes its members 1 roster notation is a similar notation used for specifying a set by enumerating its elements specifying sets by member properties is allowed by the axiom schema of specification this is also known as set comprehension and set abstraction sets defined by a predicate edit set builder notation can be used to describe a set that is defined by a predicate that is a logical formula that evaluates to true for an element of the set and false otherwise 2 in this form set builder notation has three parts a variable a colon or vertical bar separator and a predicate thus there is a variable on the left of the separator and a rule on the right of it these three parts are contained in curly brackets x φ x displaystyle x mid phi x or x φ x displaystyle x phi x the vertical bar or colon is a separator that can be read as such that for which or with the property that the formula φ x is said to be the rule or the predicate all values of x for which the predicate holds is true belong to the set being defined all values of x for which the predicate does not hold do not belong to the set thus x φ x displaystyle x mid phi x is the set of all values of x that satisfy the formula φ 3 it may be the empty set if no value of x satisfies the formula specifying the domain edit a domain e that is a set to which the notation defines a subset can appear on the left of the vertical bar 4 x e φ x displaystyle x in e mid phi x or by adjoining it to the predicate x x e and φ x or x x e φ x displaystyle x mid x in e text and phi x quad text or quad x mid x in e land phi x the symbol displaystyle in denotes set membership and the symbol displaystyle land denotes the logical and operator this notation represents the set of all values of x that belong to the set e and for which the predicate is true the subset axiom insures that if e is a set and φ displaystyle phi is a predicate the notation defines always a set which is a subset of e if φ x displaystyle phi x has the form φ 1 x φ 2 x displaystyle phi _ 1 x land phi _ 2 x then x e φ x displaystyle x in e mid phi x is sometimes written x e φ 1 x φ 2 x displaystyle x in e mid phi _ 1 x phi _ 2 x using a comma instead of the symbol displaystyle land in cases where the set e is clear from context it may be not explicitly specified it is common in the literature for an author to state the domain ahead of time and then not specify it in the set builder notation for example an author may say something such as unless otherwise stated variables are to be taken to be natural numbers though in less formal contexts where the domain can be assumed a written mention is often unnecessary in the cases where the domain e is not defined implicitly or explicitly it may occur that the notation does not define a set but a proper class 5 for example russell s paradox shows that the expression x x x displaystyle x mid x not in x although seemingly well formed as a set builder expression cannot define a set without producing a contradiction 6 in some cases the notation may define a set that is not a subset of a previously defined set this is the case of the power set of a set e which is defined as p e x x e x y x y e displaystyle mathcal p e x mid x subset e x mid y in x implies y in e the fact that this defines a set is an axiom of set theory called the axiom of power set examples edit the following examples illustrate particular sets defined by set builder notation via predicates in each case the domain is specified on the left side of the vertical bar while the rule is specified on the right side x r x 0 displaystyle x in mathbb r mid x 0 is the set of all strictly positive real numbers which can be written in interval notation as 0 displaystyle 0 infty x r x 1 displaystyle x in mathbb r mid x 1 is the set 1 1 displaystyle 1 1 this set can also be defined as x r x 2 1 displaystyle x in mathbb r mid x 2 1 see equivalent predicates yield equal sets below for each integer m we can define g m x z x m m m 1 m 2 displaystyle g_ m x in mathbb z mid x geq m m m 1 m 2 ldots as an example g 3 x z x 3 3 4 5 displaystyle g_ 3 x in mathbb z mid x geq 3 3 4 5 ldots and g 2 2 1 0 displaystyle g_ 2 2 1 0 ldots x y r r 0 y f x displaystyle x y in mathbb r times mathbb r mid 0 y f x is the set of pairs of real numbers such that y is greater than 0 and less than f x for a given function f here the cartesian product r r displaystyle mathbb r times mathbb r denotes the set of ordered pairs of real numbers n n k k n n 2 k displaystyle n in mathbb n mid exists k k in mathbb n land n 2k is the set of all even natural numbers the displaystyle land sign stands for and which is known as logical conjunction the sign stands for there exists which is known as existential quantification so for example x p x displaystyle exists x p x is read as there exists an x such that p x n k n n 2 k displaystyle n mid exists k in mathbb n n 2k is a notational variant for the same set of even natural numbers it is not necessary to specify that n is a natural number as this is implied by the formula on the right a r p z q z q 0 a q p displaystyle a in mathbb r mid exists p in mathbb z exists q in mathbb z q not 0 land aq p is the set of rational numbers that is real numbers that can be written as the ratio of two integers more complex expressions on the left side of the notation edit an extension of set builder notation replaces the single variable x with an expression so instead of x φ x displaystyle x mid phi x we may have f x φ x displaystyle f x mid phi x which should be read f x φ x y x y f x φ x displaystyle f x mid phi x y mid exists x y f x wedge phi x for example 2 n n n displaystyle 2n mid n in mathbb n where n displaystyle mathbb n is the set of all natural numbers is the set of all even natural numbers p q p q z q 0 displaystyle p q mid p q in mathbb z q not 0 where z displaystyle mathbb z is the set of all integers is q displaystyle mathbb q the set of all rational numbers 2 t 1 t z displaystyle 2t 1 mid t in mathbb z is the set of odd integers t 2 t 1 t z displaystyle t 2t 1 mid t in mathbb z creates a set of pairs where each pair puts an integer into correspondence with an odd integer when inverse functions can be explicitly stated the expression on the left can be eliminated through simple substitution consider the example set 2 t 1 t z displaystyle 2t 1 mid t in mathbb z make the substitution u 2 t 1 displaystyle u 2t 1 which is to say t u 1 2 displaystyle t u 1 2 then replace t in the set builder notation to find 2 t 1 t z u u 1 2 z displaystyle 2t 1 mid t in mathbb z u mid u 1 2 in mathbb z equivalent predicates yield equal sets edit two sets are equal if and only if they have the same elements sets defined by set builder notation are equal if and only if their set builder rules including the domain specifiers are equivalent that is x a p x x b q x displaystyle x in a mid p x x in b mid q x if and only if t t a p t t b q t displaystyle forall t t in a land p t leftrightarrow t in b land q t therefore in order to prove the equality of two sets defined by set builder notation it suffices to prove the equivalence of their predicates including the domain qualifiers for example x r x 2 1 x q x 1 displaystyle x in mathbb r mid x 2 1 x in mathbb q mid x 1 because the two rule predicates are logically equivalent x r x 2 1 x q x 1 displaystyle x in mathbb r land x 2 1 leftrightarrow x in mathbb q land x 1 this equivalence holds because for any real number x we have x 2 1 displaystyle x 2 1 if and only if x is a rational number with x 1 displaystyle x 1 in particular both sets are equal to the set 1 1 displaystyle 1 1 set existence axiom edit in many formal set theories such as zermelo fraenkel set theory set builder notation is not part of the formal syntax of the theory instead there is a set existence axiom scheme which states that if e is a set and φ x is a formula in the language of set theory then there is a set y whose members are exactly the elements of e that satisfy φ e y x x y x e φ x displaystyle forall e exists y forall x x in y leftrightarrow x in e land phi x the set y obtained from this axiom is exactly the set described in set builder notation as x e φ x displaystyle x in e mid phi x in programming languages edit main article list comprehension a similar notation available in a number of programming languages notably python and haskell is the list comprehension which combines map and filter operations over one or more lists it has been suggested that parts of this page be moved into list comprehension discuss december 2023 in python the set builder s braces are replaced with square brackets parentheses or curly braces giving list generator and set objects respectively python uses an english based syntax haskell replaces the set builder s braces with square brackets and uses symbols including the standard set builder vertical bar the same can be achieved in scala using sequence comprehensions where the for keyword returns a list of the yielded variables using the yield keyword 7 consider these set builder notation examples in some programming languages example 1 example 2 set builder l l l displaystyle l l in l k x k k x x p x displaystyle k x k in k wedge x in x wedge p x python l for l in l k x for k in k for x in x if p x haskell l l ls k x k ks x xs p x scala for l l yield l for k k x x if p x yield k x c from l in l select l from k in k from x in x where p x select k x sql select l from l_set select k x from k_set x_set where p x prolog setof l member l ls result setof k x member k ks member x xs call p x result erlang l l ls k x k ks x xs p x julia l for l l k x for k k for x x if p x mathematica l l l cases tuples k x k_ x_ p x tuples k select x p the set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions which permits map filter like operations over any monad with a zero element see also edit glossary of set theory notes edit rosen kenneth 2007 discrete mathematics and its applications 6th ed new york ny mcgraw hill pp 111 112 isbn 978 0 07 288008 3 michael j cullinan 2012 a transition to mathematics with proofs jones bartlett pp 44ff weisstein eric w set mathworld wolfram com retrieved 20 august 2020 set builder notation mathsisfun com retrieved 20 august 2020 richard s pierce 1968 introduction to the theory of abstract algebras page 2 irvine andrew david deutsch harry 9 october 2016 1995 russell s paradox stanford encyclopedia of philosophy retrieved 6 august 2017 sequence comprehensions scala retrieved 6 august 2017 v t e set theory overview set mathematics axioms adjunction choice countable dependent global constructibility v l determinacy projective extensionality infinity limitation of size pairing power set regularity union martin s axiom axiom schema replacement specification operations cartesian product complement i e set difference de morgan s laws disjoint union identities intersection power set symmetric difference union concepts methods almost cardinality cardinal number large class constructible universe continuum hypothesis diagonal argument element ordered pair tuple family forcing one to one correspondence ordinal number set builder notation transfinite induction venn diagram set types amorphous countable empty finite hereditarily filter base subbase ultrafilter fuzzy infinite dedekind infinite recursive singleton subset superset transitive uncountable universal theories alternative axiomatic constructive naive cantor s theorem zermelo general principia mathematica new foundations zermelo fraenkel von neumann bernays gödel morse kelley kripke platek tarski grothendieck paradoxes problems russell s paradox suslin s problem burali forti paradox set theorists paul bernays georg cantor paul cohen richard dedekind abraham fraenkel kurt gödel thomas jech john von neumann willard quine bertrand russell thoralf skolem ernst zermelo retrieved from https en wikipedia org w index php title set builder_notation oldid 1368739220 categories set theory mathematical notation hidden categories articles with short description short description is different from wikidata use dmy dates from december 2020 articles proposed for splitting from december 2023 all articles proposed for splitting articles proposed for section moving from december 2023 all articles proposed for section moving articles with example haskell code articles with example python programming language code this page was last edited on 10 august 2026 at 20 35 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents set builder notation 10 languages add topic