Power Set - Properties

Properties

If S is a finite set with |S| = n elements, then the number of subsets of S is . This fact, which is the motivation for the notation 2S, may be demonstrated by a simple inductive argument that illuminates the power set's combinatorial structure:

If n is zero, then S is the empty set, which has exactly subset (namely itself). Otherwise, n is some positive number, in which case one can select any element x of S and gather up the subsets of S itself into pairs that are identical excepting that does not include x and does. Thus, removing x from S would result in a new set whose own powerset—containing every and nothing else—is exactly half the size of . One can view S as having been built up from the empty set through successively restoring each of its n elements, and during such a construction the power set undergoes n doublings in size. Hence .

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).

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.