Power set
The elements of the power set of {x, y, z} ordered with respect to inclusion. |
| Set operation |
| Set theory |
| The power set is the set that contains all subsets of a given set. |
In mathematics, the power set [or powerset] of a set S is the set of all subsets of S, including the empty set and S itself.[1] In axiomatic set theory [as developed, for example, in the ZFC axioms], the existence of the power set of any set is postulated by the axiom of power set.[2] The powerset of S is variously denoted as P[S], 𝒫[S], P[S], , or 2S. The notation 2S, 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, equivalent to, or bijective to the set of all the functions from S to the given two elements set.[1]
Any subset of P[S] is called a family of sets over S.
Example[edit]
If S is the set {x, y, z}, then all the subsets of S are
and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.[3]
Properties[edit]
If S is a finite set with the cardinality |S| = n [i.e., the number of all elements in 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 of the notation 2S denoting the power set P[S] are demonstrated in the below.
Cantor's diagonal argument shows that the power set of a set [whether infinite or not] always has strictly higher cardinality than the set itself [or informally, the power set must be larger than the original set]. In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. The power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers [see Cardinality of the continuum].
The power set of a set S, together with the operations of union, intersection and complement, can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras, this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra [see Stone's representation theorem].
The power set of a set S forms an abelian group 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. 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, XY 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], 2S [i.e., {0,1}S] is the set of all functions from S to {0,1}. As shown above, 2S and the power set of S, P[S], is 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 2n − 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:
| { } | 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 a bijective 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 bijective 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 using the formula:
Therefore, one can deduce the following identity, assuming :
Recursive definition[edit]
If is a finite set, then a recursive definition of proceeds as follows:
In words:
Subsets of limited cardinality[edit]
The set of subsets of S of cardinality less than or equal to κ is sometimes denoted by Pκ[S] or [S]κ, and the set of subsets with cardinality strictly less than κ is sometimes denoted P< κ[S] or [S]