Power set

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In mathematics, the power set (or powerset) of any set S, written \mathcal{P}(S), P(S), (S) or 2S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed e.g. in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.

Any subset F of \mathcal{P}(S) is called a family of sets over S.

Contents

Example

If S is the set {x, y, z}, then the subsets of S are:

  • {} (also denoted \varnothing, the empty set)
  • {x}
  • {y}
  • {z}
  • {x, y}
  • {x, z}
  • {y, z}
  • {x, y, z}

and hence the power set of S is

Properties

If S is a finite set with |S| = n elements, then the power set of S contains |\mathcal{P}(S)| = 2^n elements.[1]

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 (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. For example, 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).

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