Power set

In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S), ?(S), ℘(S) (using the 'Weierstrass p'), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. In mathematics, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as P(S), ?(S), ℘(S) (using the 'Weierstrass p'), P(S), ℙ(S), or, identifying the powerset of S with the set of all functions from S to a given set of two elements, 2S. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. Any subset of P(S) is called a family of sets over S. If S is the set {x, y, z}, then the subsets of S are and hence the power set of S is {{},{x},{y},{z},{x, y},{x, z},{y, z},{x, y, z}}. If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2n. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows, 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. 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). 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 can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem). The power set of a set S forms an abelian group when considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse) and a commutative monoid 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 Boolean ring. 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 P(S), where each function is the characteristic function of the subset in P(S) with which it is identified. Hence 2S and P(S) could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2S: the fact that this function-representation of subsets makes it a special case of the XY notation and the property, mentioned above, that |2S| = 2|S|.)