Power set
The set of all subsets of a given set.
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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. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The power set is a fundamental concept in set theory, with applications in Boolean algebra, measure theory, and combinatorics.
- definition
- Set of all subsets of a given set
- notation
- P(S), 𝒫(S), P(S), ℙ(S), or 2^S
- cardinality_finite
- If |S| = n, then |P(S)| = 2^n
- key_theorem
- Cantor's diagonal argument shows power set has strictly higher cardinality than the original set
- algebraic_structure
- Forms a Boolean algebra, an abelian group under symmetric difference, and a commutative monoid under intersection
- example
- For S = {x, y, z}, P(S) has 8 subsets
Lore & Background
The power set of a set S is defined as the collection of all subsets of S, including the empty set and S itself. For a finite set with n elements, the power set contains exactly 2^n subsets, a fact that motivates the notation 2^S. This equivalence is demonstrated through indicator functions: each subset corresponds to a function from S to {0,1}, and the set of all such functions is denoted {0,1}^S, which is bijective to the power set.
Reader's Guide
The power set is central to set theory and its applications. Cantor's theorem shows that the power set of any set has a strictly larger cardinality than the set itself, proving that there is no largest set and establishing the existence of multiple infinities. The power set of the natural numbers has the same cardinality as the real numbers. In algebra, the power set forms a Boolean algebra under union, intersection, and complement, and any finite Boolean algebra is isomorphic to the power set of a finite set. The power set also forms an abelian group under symmetric difference and a commutative monoid under intersection, together forming a Boolean ring. The recursive definition of the power set provides a constructive method for generating all subsets.
Did You Know?
- The power set of a set with n elements has exactly 2^n subsets.
- Cantor's diagonal argument shows the power set always has strictly higher cardinality than the original set.
- The power set of the set of natural numbers can be put in one-to-one correspondence with the set of real numbers.
- The power set of a set S, with union, intersection, and complement, is a prototypical example of a Boolean algebra.
Frequently Asked Questions
What is the Power set?
The Power set of a set S is the collection containing every possible subset of S, which includes both the empty set and S itself. It is the foundational 'set of all subsets' concept in axiomatic set theory, guaranteed to exist by the axiom of power set.
How do you write or denote the Power set?
You will see it rendered as P(S), 𝒫(S), ℙ(S), or even 2^S, with the script P being the most common in formal texts. The 2^S notation is a hint at the exponential relationship between the size of S and the size of its power set.
If a set has n elements, how many subsets does its Power set contain?
The power set of an n-element set always has exactly 2^n elements. For example, a three-element set like {x, y, z} yields a power set of eight distinct subsets.
Why is Cantor's diagonal argument tied to the Power set?
Cantor's diagonal argument proves that no set can be placed in one-to-one correspondence with its own power set, meaning the power set always has strictly greater cardinality. This result is the engine behind the fact that there is no single 'largest' infinity.
What algebraic structures does the Power set support?
Equipped with union and intersection, the power set forms a Boolean algebra; under symmetric difference it becomes an abelian group; and under intersection alone it acts as a commutative monoid. These structures make it a workhorse in Boolean algebra, measure theory, and combinatorics.
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