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Power set

In , the power set of a set S, denoted \mathcal{P}(S) or $2^S, is the set of all subsets of S, including the \emptyset and S itself. For a S with n, the power set has $2^n. The concept of the power set emerged in the late as part of Georg Cantor's pioneering work on and transfinite numbers, where he explored operations on point sets and their subsets starting around 1872. In modern axiomatic , specifically with the (ZFC), the Axiom of the Power Set asserts that for every set x, there exists a set y (the power set) such that every of x is an element of y; formally, \forall x \exists y (\forall z (z \in y \leftrightarrow z \subseteq x)). A key result associated with power sets is , which states that for any set S, the cardinality of \mathcal{P}(S) is strictly greater than the cardinality of S, implying that there is no largest infinite cardinal and establishing a hierarchy of infinities. This theorem, first proved by Cantor in 1891 using a diagonal argument, has profound implications for understanding uncountable sets and leads to paradoxes like when considering the "set of all sets."

Definition and Notation

Formal Definition

In set theory, a subset A of a set S, denoted A \subseteq S, is a set such that every element of A is also an element of S. The power set of a set S, denoted \mathcal{P}(S) or P(S), is the set whose elements are all possible of S. By definition, this includes the \emptyset (the containing no elements) and S itself (the containing all elements of S) as members of \mathcal{P}(S). For a S with |S| = n, the power set \mathcal{P}(S) contains exactly $2^n elements. The concept of the power set was developed by in the late 19th century as part of the foundations of .

Standard Notation

The power set of a set S is conventionally denoted by P(S), where P stands for "power," indicating the collection of all subsets of S. An alternative notation is \mathcal{P}(S), using a script or calligraphic P to distinguish it in formal mathematical writing. Another common symbol is $2^S, which emphasizes the in the number of subsets relative to the size of S. The notation $2^S originates from the bijection between the power set \mathcal{P}(S) and the set of all functions from S to \{0,1\}, where each function corresponds to a characteristic function of a subset: for an element x \in S, the value 1 indicates inclusion in the subset, and 0 indicates exclusion. Here, 2 is understood as the two-element set \{0,1\}, and $2^S denotes the set of functions with domain S and codomain \{0,1\}. In set theory, the subset relation is standardly denoted by \subseteq, so A \subseteq B means every element of A belongs to B. A proper subset, where A \subseteq B but A \neq B, is often denoted by \subset, though conventions vary: some texts reserve \subset for the non-strict case and use \subsetneq or \varsubsetneq for proper subsets. These symbols appear consistently across mathematical literature, with \subseteq and its dual \supseteq being the most widely adopted for inclusivity. Variations in notation occur across fields; for instance, in , subset relations may align with entailment symbols like \vdash, but set-theoretic conventions predominate in core usage.

Examples and Illustrations

Finite Set Examples

To illustrate the power set of a , consider the simplest case where S = \{1\}. The power set P(S) consists of the and the set itself, so P(S) = \{\emptyset, \{1\}\}. This yields 2 subsets. For a set with two elements, let S = \{1, 2\}. The subsets include the , the singletons, and the full set, giving P(S) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}. There are 4 subsets in total. A more detailed example arises with S = \{a, b, c\}, which has 8 subsets. These can be systematically listed as follows:
Subset
\emptyset
\{a\}
\{b\}
\{c\}
\{a, b\}
\{a, c\}
\{b, c\}
\{a, b, c\}
This enumeration confirms the pattern that adding each new element to a finite set doubles the number of subsets in its power set.

Infinite Set Examples

The power set of the natural numbers, denoted P(\mathbb{N}), comprises all subsets of \mathbb{N}, encompassing both finite subsets such as \{1, 3\} and infinite subsets such as the even natural numbers \{2, 4, 6, \dots \}. These subsets illustrate the diversity within P(\mathbb{N}), which includes collections of arbitrary size and structure, but unlike finite cases where all subsets can be explicitly listed, the elements of P(\mathbb{N}) cannot be exhaustively enumerated due to its uncountable nature. A fundamental result in set theory, known as Cantor's theorem, establishes that for any set S, the cardinality of its power set satisfies |P(S)| > |S|. Applied to \mathbb{N}, where |\mathbb{N}| = \aleph_0, this implies |P(\mathbb{N})| = 2^{\aleph_0}, the cardinality of the continuum, highlighting the explosive growth in size when forming power sets from infinite domains. Similarly, the power set of the real numbers, P(\mathbb{R}), demonstrates this scale on an even grander level. With |\mathbb{R}| = 2^{\aleph_0}, Cantor's theorem yields |P(\mathbb{R})| = 2^{2^{\aleph_0}}, a cardinal vastly exceeding the continuum itself. Examples of subsets in P(\mathbb{R}) include the rational numbers \mathbb{Q}, which form a countable dense subset, or open intervals such as (0, 1), but these represent only a minuscule fraction of the unlistable totality within P(\mathbb{R}).

Cardinality and Size

Cardinality Formula

The cardinality of the power set P(S) of any set S is given by the formula |P(S)| = 2^{|S|}, where $2^{|S|} denotes the of the set of all functions from S to the set \{0,1\}. For a finite set S with |S| = n, this formula yields |P(S)| = 2^n. A proof proceeds by on n. The base case n = 0 holds since P(\emptyset) = \{\emptyset\}, which has 1 = $2^0. For the inductive step, assume the result for n = k, so |P(T)| = 2^k for any set T with k elements. Now let S have k+1 elements, say S = T \cup \{x\} where x \notin T and |T| = k. The subsets of S into those not containing x (isomorphic to P(T), $2^k) and those containing x (isomorphic to P(T), another $2^k); thus, |P(S)| = 2 \cdot 2^k = 2^{k+1}. This cardinality arises from a natural between P(S) and the set of sequences of n, or equivalently, the set of functions \chi_A: S \to \{0,1\} for subsets A \subseteq S, where \chi_A(y) = 1 if y \in A and 0 otherwise. Each of the n elements of S has 2 choices (in or out of the subset), yielding $2^n possibilities./01:_Set_Theory/1.03:_Cartesian_Products_and_Power_Sets) The formula extends to sets via arithmetic, where for a \kappa = |S|, the $2^\kappa is defined as the of the power set P(\kappa) of a set of \kappa, or equivalently, the set \{0,1\}^\kappa of functions from a set of \kappa to \{0,1\}. The via functions preserves this for S.

Finite vs. Infinite Cases

For finite sets, the cardinality of the power set \mathcal{P}(S) of a set S with n elements is exactly $2^n, providing a precise and computable measure of its size. However, despite this exact computability, enumerating all elements of \mathcal{P}(S) becomes impractical for large n due to the exponential growth, requiring time proportional to $2^n in the worst case, which exceeds polynomial-time bounds. In contrast, for any S, asserts a strict in cardinalities: |S| < |\mathcal{P}(S)|, ensuring the power set is always larger. This result is established via , which constructs a of S not in any purported surjection from S to \mathcal{P}(S), proving no such surjection exists. For the countable infinity of the natural numbers, with cardinality \aleph_0, the power set achieves cardinality $2^{\aleph_0}, the size of the . These distinctions carry profound implications: finite power sets remain fully manageable in theory, albeit computationally burdensome for substantial n, while infinite power sets engender a transfinite through , where each successive power set yields a larger , underpinning the existence of uncountably many . The uncountability introduced by infinite power sets forms a cornerstone of , enabling the study of continuous structures like the real line, and of , where it highlights limitations in enumeration and decidability./08%3A_Cardinality/8.03%3A_Cantors_Theorem)

Algebraic Properties

Closure Under Operations

The power set of a set S, denoted \mathcal{P}(S), exhibits under the fundamental set operations of , , and relative complement. Specifically, if A \subseteq S and B \subseteq S, then A \cup B \subseteq S, ensuring that the of any two s remains a of S and thus belongs to \mathcal{P}(S). Similarly, A \cap B \subseteq S, so the is also an element of \mathcal{P}(S). For the relative complement, given A \subseteq S, the set S \setminus A consists of elements in S but not in A, which is clearly a of S, hence S \setminus A \in \mathcal{P}(S). This closure property extends to finite and countable combinations of these operations, as repeated applications keep results within subsets of S. The structure (\mathcal{P}(S), \cup, \cap, \setminus, \emptyset, S) forms a , where serves as the join operation, as the meet, relative complement as the negation, the \emptyset as the , and S as the unit element. In this algebra, every element has a unique complement, and the operations satisfy the axioms of algebras, including complementarity and . Furthermore, the Boolean algebra structure of \mathcal{P}(S) ensures distributivity of the operations. For instance, union distributes over intersection: for any A, B, C \subseteq S, A \cup (B \cap C) = (A \cup B) \cap (A \cup C), and dually, intersection distributes over union: A \cap (B \cup C) = (A \cap B) \cup (A \cap C). These distributive laws hold because they are inherited from the corresponding properties of set operations on subsets of S.

Intersection and Union Properties

In the power set \mathcal{P}(S) of a set S, the operations of union and intersection satisfy the idempotence property. For any subset A \subseteq S, it holds that A \cup A = A and A \cap A = A. This reflects the fact that including an element already present in A does not alter the set, mirroring the idempotent nature of logical disjunction and conjunction in the underlying Boolean structure of \mathcal{P}(S). Absorption laws further characterize the interactions between and within \mathcal{P}(S). Specifically, for subsets A, B \subseteq S, A \cup (A \cap B) = A and A \cap (A \cup B) = A. These identities arise because the A \cap B is contained in A, so unioning it with A yields A, while the A \cup B contains A, so intersecting it with A also yields A. Such properties underscore the algebraic closure of \mathcal{P}(S) under these operations, facilitating simplifications in set-theoretic expressions. De Morgan's laws provide duality between union, intersection, and complementation in \mathcal{P}(S), where the complement of a subset A \subseteq S is A^c = S \setminus A. For any A, B \subseteq S, (A \cup B)^c = A^c \cap B^c and (A \cap B)^c = A^c \cup B^c. These laws, which interchange union and intersection under complement, are foundational to the Boolean algebra structure of the power set and enable transformations between disjunctive and conjunctive forms in logical and set contexts. The power set \mathcal{P}(S) is closed under arbitrary unions and intersections of its elements. For any family of subsets \{A_i\}_{i \in I} with each A_i \subseteq S, the \bigcup_{i \in I} A_i \subseteq S and thus belongs to \mathcal{P}(S), regardless of whether the I is finite or . Similarly, \bigcap_{i \in I} A_i \subseteq S is in \mathcal{P}(S). This holds even for families, as the resulting sets remain subsets of S, though in axiomatic frameworks, the of such unions relies on the . For finite S, families may redundantly cover subsets, but the property persists.

Representations of Subsets

Functional Representation

The functional representation of subsets of a set S employs , which map elements of S to the binary set \{0, 1\}. For any A \subseteq S, the \chi_A: S \to \{0, 1\} is defined such that \chi_A(x) = 1 if x \in A and \chi_A(x) = 0 otherwise. This function serves as an indicator of membership in A, providing a precise encoding of the subset's structure. As an illustration, consider the set S = \{1, 2\}. The for the A = \{1\} is given by \chi_A(1) = 1 and \chi_A(2) = 0, which can be denoted as the (1, 0) when S is enumerated. Similarly, the empty \emptyset corresponds to the constant \chi_\emptyset(x) = 0 for all x \in S, while the full set S corresponds to \chi_S(x) = 1 for all x \in S. This representation extends naturally to infinite sets, where the assigns binary values without requiring an explicit . The collection of all such characteristic functions establishes a bijection between the power set \mathcal{P}(S) and the function set \{0,1\}^S. Specifically, the mapping A \mapsto \chi_A is injective because if \chi_A = \chi_B, then A = B as the preimage of 1 under \chi_A is exactly A; it is surjective since any function f: S \to \{0,1\} equals \chi_B where B = f^{-1}(\{1\}). This isomorphism highlights the structural equivalence between subsets and binary-valued functions. A key benefit of this functional perspective is that it translates the partial order of subset inclusion into the pointwise order on functions: A \subseteq B if and only if \chi_A(x) \leq \chi_B(x) for every x \in S, preserving the relational structure of the power set in a functional framework.

Binary String Representation

For a finite set S = \{x_1, x_2, \dots, x_n\} with a fixed ordering of its elements, each A \subseteq S can be represented by a binary string of n, where the i-th bit b_i is 1 if x_i \in A and 0 otherwise. This encoding, often called a bit string or bitmask representation, directly maps the power set \mathcal{P}(S) to the set of all possible strings of length n, providing a compact way to enumerate or manipulate subsets. For example, consider S = \{a, b, c\}. The subset A = \{a, c\} corresponds to the binary string 101, indicating inclusion of the first and third elements while excluding the second. The empty subset \emptyset maps to 000, and the full set S to 111. This representation depends on the chosen ordering of S; different enumerations yield different binary strings for the same subsets, though the bijection between subsets and strings remains one-to-one. In applications, representations enable efficient storage and operations on subsets, such as bitwise AND for and OR for , which run in O(n) time. They are particularly useful in dynamic programming algorithms, like those solving the , where bitmasks index states representing partial subsets to track achievable sums.

Connections to Combinatorics

Relation to Binomial Coefficients

The provides a fundamental algebraic connection between the power set of a and . It states that for any nonnegative n and real numbers x and y, (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k, where \binom{n}{k} denotes the , representing the number of ways to choose k elements from n. A special case arises when x = 1 and y = 1, yielding (1 + 1)^n = 2^n = \sum_{k=0}^n \binom{n}{k}. This equality links the of the power set |\mathcal{P}(S)| = 2^n for a S with |S| = n directly to the sum of binomial coefficients. Combinatorially, each \binom{n}{k} counts the subsets of S with exactly k elements, so the sum \sum_{k=0}^n \binom{n}{k} enumerates all possible subsets, confirming the total number is $2^n. This summation also interprets the power set size through generating functions, where the polynomial (1 + x)^n = \sum_{k=0}^n \binom{n}{k} x^k has coefficients that count subsets by size, and evaluating at x = 1 gives the full cardinality $2^n./08:_Combinatorics/8.05:_The_Binomial_Theorem)

Subsets of Fixed Cardinality

In , the subsets of a given set S with exactly k elements are known as the k-subsets of S. The collection of all such k-subsets forms a set, often denoted \binom{S}{k} or C(S, k), which can be formally defined as \{ A \subseteq S : |A| = k \}. For a finite set S with |S| = n, the cardinality of C(S, k) is given by the binomial coefficient \binom{n}{k}, representing the number of ways to choose k elements from n without regard to order. This count arises naturally in combinatorial enumeration, where each k-subset corresponds to a unique . A key property of these collections is that the power set \mathcal{P}(S) of a S with |S| = n is the disjoint union of the sets C(S, k) for k = 0, 1, \dots, n. That is, \mathcal{P}(S) = \bigsqcup_{k=0}^{n} C(S, k), and the total number of subsets satisfies |\mathcal{P}(S)| = \sum_{k=0}^{n} \binom{n}{k} = 2^n. Additionally, the binomial coefficients exhibit symmetry: \binom{n}{k} = \binom{n}{n-k}, which reflects the equivalence between selecting k elements to include and n-k elements to exclude from S. These k-subsets find applications in various combinatorial problems, such as selecting from a group of , where the of selection does not matter and the is fixed at k. For instance, the number of possible of 3 from 10 candidates is \binom{10}{3} = 120.

Recursive and Constructive Aspects

Recursive Construction

The of a set can be constructed recursively by starting with the and iteratively adding , deciding for each new whether to include it in the subsets or not. The base case is the of the , defined as P(\emptyset) = \{\emptyset\}. For a set S and an element x \notin S, the recursive step extends the as follows: P(S \cup \{x\}) = P(S) \cup \{ A \cup \{x\} \mid A \in P(S) \}. This construction ensures that every of S \cup \{x\} is either a of S (from P(S)) or a of S with x added (from the second term). The two collections in the are disjoint because no in P(S) contains x, while every set in the second collection does. This recursive process highlights the conceptual building of the power set: at each step, for the new x, one doubles the existing subsets by choosing to include or exclude x, mirroring the decision for membership. Consequently, the satisfies |P(S \cup \{x\})| = 2 |P(S)|, as the two parts have equal size |P(S)|. To establish that the power set of a S with |S| = n has $2^n, proceed by on n. For the base case n=[0](/page/0), |P(\emptyset)| = [1](/page/1) = 2^[0](/page/0). Assume the holds for some k = |S|; then for S' = S \cup \{x\} with x \notin S, |P(S')| = 2 |P(S)| = 2 \cdot 2^k = 2^{k+1}, completing the inductive step. Thus, |P(S)| = 2^n for S.

Iterative Building Process

The iterative building process for the power set of a finite set provides a constructive method to generate all subsets by incrementally incorporating each element. This algorithm starts with the power set of the empty set, which is P(∅) = {∅}. For each subsequent element x added to the current set S, the existing power set P(S) is duplicated, and x is appended to every subset in the duplicate to form new subsets; the original P(S) and the modified duplicates are then united to yield P(S ∪ {x}). This mirrors the mathematical relation P(S ∪ {x}) = P(S) ∪ {T ∪ {x} \mid T \in P(S)}, where the two collections are disjoint since x \notin S. To illustrate, consider constructing the power set for S = {1, 2, 3} step by step:
  • Initialize: P(∅) = { ∅ }
  • Add 1: Duplicate { ∅ } to get { ∅ }, append 1 to get { {1} }; union yields { ∅, {1} }
  • Add 2: Duplicate { ∅, {1} } to get { ∅, {1} }, append 2 to get { {2}, {1,2} }; union yields { ∅, {1}, {2}, {1,2} }
  • Add 3: Duplicate { ∅, {1}, {2}, {1,2} } to get { ∅, {1}, {2}, {1,2} }, append 3 to get { {3}, {1,3}, {2,3}, {1,2,3} }; union yields { ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }
This process demonstrates how the power set grows by doubling in size with each added element, resulting in $2^{|S|} subsets overall. The of this iterative construction is O(n 2^n), where n = |S|, due to generating 2^n subsets with an average cost per subset for construction and copying. In programming contexts, this method is commonly implemented for subset generation algorithms, such as those solving the subsets problem or enumerating combinations in .

Categorical and Logical Perspectives

Power Object in Category Theory

In category theory, particularly within categories with finite limits such as the \mathbf{Set}, the power object P(S) of an object S is an object equipped with a \in_S : P(S) \times S \to \Omega representing the membership relation, where \Omega is the subobject classifier (in \mathbf{Set}, \Omega = 2 = \{0,1\}). This structure generalizes the classical power set by abstracting subsets to subobjects classified via characteristic morphisms. The universal property of the power object states that for any object B and (subobject) R \hookrightarrow B \times S, there exists a unique characteristic \chi_R : B \to P(S) such that R is the pullback of \in_S along \mathrm{id}_S \times \chi_R. This bijection between relations into S and morphisms into P(S) mirrors how subsets are classified by their characteristic functions in . In the \mathbf{Set}, the power object P(S) is isomorphic to the power set \mathcal{P}(S), with the membership given by (A, s) \mapsto 1 if s \in A and $0 otherwise, corresponding to characteristic functions S \to 2. This construction generalizes beyond \mathbf{Set} to any topos—a category with finite limits, exponentials, and a subobject classifier—where power objects exist for every object and play a foundational role in higher category theory by enabling internal logic and geometric interpretations of sets. The existence of power objects in a topos ensures the category supports a rich structure for reasoning about subobjects categorically, influencing applications in synthetic differential geometry and type theory.

Role in Quantifiers and Functors

In mathematical logic, the power set plays a fundamental role in expressing restricted quantifiers through subset relations. The universal quantifier restricted to a set S, denoted \forall x \in S \, \phi(x), is logically equivalent to the statement that S \subseteq \{ x \mid \phi(x) \}, where \{ x \mid \phi(x) \} represents the definable subset of elements satisfying the predicate \phi. This equivalence relies on the power set \mathcal{P}(U) of the universe U, as it encompasses all possible subsets, including those definable by formulas like \phi, allowing the representation of quantified statements as membership in subsets. Similarly, the existential quantifier restricted to S, \exists x \in S \, \phi(x), holds the definable set \{ x \mid \phi(x) \} intersects non-emptily with S, i.e., \{ x \mid \phi(x) \} \cap S \neq \emptyset. Here, the power set facilitates the construction of such definable subsets within a larger ambient set, enabling the translation of existential claims into set-theoretic operations on subsets. This approach underscores how power sets provide the structural foundation for interpreting restricted quantification in over sets. In functorial contexts, the power set operation defines the covariant functor \mathcal{P}: \mathbf{Set} \to \mathbf{Set}, which maps each set S to its power set \mathcal{P}(S) and each f: S \to T to the direct image map f_*: \mathcal{P}(S) \to \mathcal{P}(T) given by f_*(A) = \{ f(a) \mid a \in A \} for A \subseteq S. Although covariant overall, this functor exhibits contravariant behavior on hom-sets, as preimages under f reverse inclusions. The power set also induces a contravariant functor \mathcal{P}: \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}, mapping f: S \to T to the preimage map f^*: \mathcal{P}(T) \to \mathcal{P}(S) defined by f^*(B) = f^{-1}(B) for B \subseteq T. These functors preserve certain categorical structures, such as the power set functor maintaining colimits in specific filtered or finite contexts within the . Within , power sets model the by providing the full collection of subsets from which definable ones can be extracted via the of separation. The power set ensures the existence of \mathcal{P}(A) for any set A, containing all subsets B \subseteq A, while the allows the formation of \{ x \in A \mid \phi(x) \} for any \phi, directly tying into the subset representations used in quantified logic. This interplay validates the comprehension principle for definable subsets, grounding logical quantifiers in the set-theoretic framework.