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when it is considered with the operation of symmetric difference with the empty set as the identity element and each set being its own inverse and a commutative monoid when considered with the operation of intersection with the entire set s as the identity element it can hence be shown by proving the distributive laws that the power set considered together with both of these operations forms a boolean ring representing subsets as functions edit in set theory x y is the notation representing the set of all functions from y to x as 2 can be defined as 0 1 see for example von neumann ordinals 2 s i e 0 1 s is the set of all functions from s to 0 1 as shown above 2 s and the power set of s p s are considered identical set theoretically this equivalence can be applied to the example above in which s x y z to get the isomorphism with the binary representations of numbers from 0 to 2 n 1 with n being the number of elements in the set s or s n first the enumerated set x 1 y 2 z 3 is defined in which the number in each ordered pair represents the position of the paired element of s in a sequence of binary digits such as x y 011 2 x of s is located at the first from the right of this sequence and y is at the second from the right and 1 in the sequence means the element of s corresponding to the position of it in the sequence exists in the subset of s for the sequence while 0 means it does not for the whole power set of s we get elements of p s where s x y z subset sequence of binary digits binary interpretation decimal equivalent 0 0 0 000 2 0 10 x 0 0 1 001 2 1 10 y 0 1 0 010 2 2 10 x y 0 1 1 011 2 3 10 z 1 0 0 100 2 4 10 x z 1 0 1 101 2 5 10 y z 1 1 0 110 2 6 10 x y z 1 1 1 111 2 7 10 such an injective mapping from p s to integers is arbitrary so this representation of all the subsets of s is not unique but the sort order of the enumerated set does not change its cardinality e g y 1 z 2 x 3 can be used to construct another injective mapping from p s to the integers without changing the number of one to one correspondences however such finite binary representation is only possible if s can be enumerated in this example x y and z are enumerated with 1 2 and 3 respectively as the position of binary digit sequences the enumeration is possible even if s has an infinite cardinality i e the number of elements in s is infinite such as the set of integers or rationals but not possible for example if s is the set of real numbers in which case we cannot enumerate all irrational numbers relation to binomial theorem edit the binomial theorem is closely related to the power set a k elements combination from some set is another name for a k elements subset so the number of combinations denoted as c n k also called binomial coefficient is a number of subsets with k elements in a set with n elements in other words it s the number of sets with k elements which are elements of the power set of a set with n elements for example the power set of a set with three elements has c 3 0 1 subset with 0 elements the empty subset c 3 1 3 subsets with 1 element the singleton subsets c 3 2 3 subsets with 2 elements the complements of the singleton subsets c 3 3 1 subset with 3 elements the original set itself using this relationship we can compute 2 s using the formula 2 s k 0 s s k displaystyle left 2 s right sum _ k 0 s binom s k therefore one can deduce the following identity assuming s n 2 s 2 n k 0 n n k displaystyle left 2 s right 2 n sum _ k 0 n binom n k recursive definition edit if s is a finite set then a recursive definition of p s proceeds as follows if s then p s otherwise let e s and t s e then p s p t t e t p t in words the power set of the empty set is a singleton whose only element is the empty set for a non empty set s let e displaystyle e be any element of the set and t its relative complement then the power set of s is a union of a power set of t and a power set of t whose each element is expanded with the e element subsets of limited cardinality edit the set of subsets of s of cardinality less than or equal to κ is sometimes denoted by p κ s or s κ and the set of subsets with cardinality strictly less than κ is sometimes denoted p κ s or s κ similarly the set of non empty subsets of s might be denoted by p 1 s or p s power object edit this section does not cite any sources please help improve this section by adding citations to reliable sources unsourced material may be challenged and removed find sources power set news newspapers books scholar jstor february 2025 learn how and when to remove this message a set can be regarded as an algebra having no nontrivial operations or defining equations from this perspective the concept of the power set of x as the set of all subsets of x generalizes naturally to the set to all subalgebras of an algebraic structure or algebra 4 5 the power set of a set when ordered by inclusion is always a complete atomic boolean algebra and every complete atomic boolean algebra arises as the lattice of all subsets of some set the generalization to arbitrary algebras is that the set of subalgebras of an algebra again ordered by inclusion is always an algebraic lattice and every algebraic lattice arises as the lattice of subalgebras of some algebra 6 so in that regard subalgebras behave analogously to subsets however there are two important properties of subsets that do not carry over to subalgebras in general first although the subsets of a set form a set as well as a lattice in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class although they can always be organized as a lattice secondly whereas the subsets of a set are in bijection with the functions from that set to the set 0 1 2 there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way certain classes of algebras enjoy both of these properties the first property is more common the case of having both is relatively rare one class that does have both is that of multigraphs given two multigraphs g and h a homomorphism h g h consists of two functions one mapping vertices to vertices and the other mapping edges to edges the set h g of homomorphisms from g to h can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set furthermore the subgraphs of a multigraph g are in bijection with the graph homomorphisms from g to the multigraph ω definable as the complete directed graph on two vertices hence four edges namely two self loops and two more edges forming a cycle augmented with a fifth edge namely a second self loop at one of the vertices we can therefore organize the subgraphs of g as the multigraph ω g called the power object of g what is special about a multigraph as an algebra is that its operations are unary a multigraph has two sorts of elements forming a set v of vertices and e of edges and has two unary operations s t e v giving the source start and target end vertices of each edge an algebra all of whose operations are unary is called a presheaf every class of presheaves contains a presheaf ω that plays the role for subalgebras that 2 plays for subsets such a class is a special case of the more general notion of elementary topos as a category that is closed and moreover cartesian closed and has an object ω called a subobject classifier although the term power object is sometimes used synonymously with exponential object y x in topos theory y is required to be ω 7 functors and quantifiers edit there is both a covariant and contravariant power set functor p set set and p set op set the covariant functor is defined more simply as the functor which sends a set s to p s and a morphism f s t here a function between sets to the image morphism that is for a x 1 x 2 p s p f a f x 1 f x 2 p t elsewhere in this article the power set was defined as the set of functions of s into the set with 2 elements formally this defines a natural isomorphism p set 2 the contravariant power set functor is different from the covariant version in that it sends f to the pre image morphism so that if f a b t p f b a this is because a general functor c c takes a morphism h a b to precomposition by h so a function h c b c c a c which takes morphisms from b to c and takes them to morphisms from a to c through b via h 8 in category theory and the theory of elementary topoi the universal quantifier can be understood as the right adjoint of a functor between power sets the inverse image functor of a function between sets likewise the existential quantifier is the left adjoint 9 see also edit cantor s theorem family of sets field of sets combination kleene star notes edit the notation 2 s meaning the set of all functions from s to a given set of two elements e g 0 1 is used because the powerset of s can be identified with is equivalent to or bijective to the set of all the functions from s to the given two element set 1 references edit 1 2 weisstein devlin 1979 p 50 puntambekar 2007 pp 1 2 bergman george m an invitation to general algebra and universal algebra pdf uc berkeley math retrieved 2025 11 23 subalgebra lattice encyclopedia of mathematics retrieved 2025 11 23 birkhoff garrett frink orrin jr 1948 representations of lattices by sets pdf transactions of the american mathematical society 64 2 299 316 doi 10 1090 s0002 9947 1948 0027263 2 cite journal cs1 maint multiple names authors list link mac lane saunders moerdijk ieke 1992 sheaves in geometry and logic a first introduction to topos theory universitext new york springer verlag isbn 978 0 387 97710 2 riehl emily 16 november 2016 category theory in context courier dover publications isbn 978 0486809038 mac lane moerdijk 1992 p 58 sfn error multiple targets 2 citerefmac_lanemoerdijk1992 help bibliography edit devlin keith j 1979 fundamentals of contemporary set theory universitext springer verlag isbn 0 387 90441 7 zbl 0407 04003 halmos paul r 1960 naive set theory the university series in undergraduate mathematics van nostrand company zbl 0087 04403 mac lane saunders moerdijk ieke 1992 sheaves in geometry and logic springer verlag isbn 0 387 97710 4 puntambekar a a 2007 theory of automata and formal languages technical publications isbn 978 81 8431 193 8 weisstein eric w power set mathworld wolfram com archived from the original on 2023 04 06 retrieved 2020 09 05 external links edit look up power set in wiktionary the free dictionary power set at planetmath power set at the n lab power object at the n lab power set algorithm in c v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics traditional classical logic logical truth tautology proposition inference logical equivalence consistency equiconsistency argument soundness validity syllogism square of opposition venn diagram propositional boolean algebra boolean functions logical connectives propositional calculus propositional formula truth tables many valued logic 3 finite predicate first order list second order monadic higher order fixed point free quantifiers predicate monadic predicate calculus set theory set hereditary class ur element ordinal number extensionality forcing relation equivalence partition set operations intersection union complement cartesian product power set identities types of sets countable uncountable empty inhabited singleton finite infinite transitive ultrafilter recursive fuzzy universal universe constructible grothendieck von neumann maps cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary theories zermelo fraenkel axiom of choice continuum hypothesis general kripke platek morse kelley naive new foundations tarski grothendieck von neumann bernays gödel ackermann constructive formal systems list language syntax alphabet arity automata axiom schema expression ground extension by definition conservative relation formation rule grammar formula atomic closed ground open free bound variable language metalanguage logical connective predicate functional variable propositional variable proof quantifier rank sentence atomic spectrum signature string substitution symbol function logical constant non logical variable term theory list example axiomatic systems list of true arithmetic peano second order elementary function primitive recursive robinson skolem of the real numbers tarski s axiomatization of boolean algebras canonical minimal axioms of geometry euclidean elements hilbert s tarski s non euclidean principia mathematica proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic automated theorem proving category theory concrete abstract category category of sets history of logic history of mathematical logic timeline logicism mathematical object philosophy of mathematics supertask mathematics portal 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 ...