Power set
inclusion. | |
Type | Set operation |
---|---|
Field | Set theory |
Statement | The power set is the set that contains all subsets of a given set. |
Symbolic statement |
In
Example
If S is the set {x, y, z}, then all the subsets of S are
- {} (also denoted or , the empty set or the null set)
- {x}
- {y}
- {z}
- {x, y}
- {x, z}
- {y, z}
- {x, y, z}
and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.^{[3]}
Properties
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)| = 2^{n}. This fact as well as the reason of the notation 2^{S} denoting the power set P(S) are demonstrated in the below.
- An von Neumann ordinals), the P(S) is also denoted as 2^{S}. Obviously |2^{S}| = 2^{|S|} holds. Generally speaking, X^{Y} is the set of all functions from Y to X and |X^{Y}| = |X|^{|Y|}.
The power set of a set S, together with the operations of
The power set of a set S forms an
Representing subsets as functions
In
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:
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
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:
Therefore, one can deduce the following identity, assuming |S| = n:
Recursive definition
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 be any element of the set and T its relative complement; then the power set of S is a unionof a power set of T and a power set of T whose each element is expanded with the e element.
Subsets of limited cardinality
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
A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective, the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra.
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
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
Functors and quantifiers
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 . 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 . The contravariant power set functor is different from the covariant version in that it sends f to the preimage morphism, so that if . This is because a general functor takes a morphism to precomposition by h, so a function , which takes morphisms from b to c and takes them to morphisms from a to c, through b via h. ^{[4]}
In
See also
Notes
- ^ 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
- ^ ^{a} ^{b} Weisstein
- ^ Devlin 1979, p. 50
- ^ Puntambekar 2007, pp. 1–2
- ISBN 978-0486809038.
- ^ Mac Lane & Moerdijk 1992, p. 58
Bibliography
- Zbl 0407.04003.
- .
- Puntambekar, A. A. (2007). Theory Of Automata And Formal Languages. Technical Publications. .
- Weisstein, Eric W. "Power Set". mathworld.wolfram.com. Archived from the original on 2023-04-06. Retrieved 2020-09-05.
External links
- Power set at PlanetMath.
- Power set at the nLab
- Power object at the nLab
- Power set Algorithm in C++