Power set

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In mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S), P(S), or 2S, is the set of all subsets of S. 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.

For example, if S is the set {a, b, c} then the complete list of subsets of S is as follows:

  • { } (the empty set)
  • {a}
  • {b}
  • {c}
  • {a, b}
  • {a, c}
  • {b, c}
  • {a, b, c}

and hence the power set of S is

\mathcal{P}(S) = \left\{\{\}, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\right\}\,\!

If S is a finite set with |S| = n elements, then the power set of S contains |\mathcal{P}(S)| = 2^n elements. (One can - and computers sometimes do - represent the elements of \mathcal{P}(S) as n-bit numbers; the n-th bit refers to presence or absence of the n-th element of S. There are 2n such numbers.)

Cantor's diagonal argument shows that the power set of a set (infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be 'greater' than the original set). The power set of the set of natural numbers for instance 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 is a subalgebra of a power set Boolean algebra (though this is not always a particularly illuminating representation of an infinite Boolean algebra).

The power set of a set S forms an Abelian group when considered with the operation of symmetric difference (with the empty set as its unit and each set being its own inverse) and a commutative semigroup 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 commutative ring.

[edit] The notations 2S and {0,1}S

In set theory, XY is the set of all functions from Y to X. As 2 can be defined as {0,1} (see natural number), 2S (i.e., {0,1}S) is the set of all functions from S to {0,1}. By identifying a function in 2S with the corresponding preimage of 1, we see that there is a bijection between 2S and \mathcal{P}(S), where each function is the characteristic function of the subset in \mathcal{P}(S) with which it is identified. Hence 2S and \mathcal{P}(S) could be considered identical set-theoretically.

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