<P> 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). </P> <P> 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). </P> <P> 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 . </P> <P> In set theory, X is the set of all functions from Y to X . As "2" can be defined as (0, 1) (see natural number), 2 (i.e., (0, 1)) is the set of all functions from S to (0, 1). By identifying a function in 2 with the corresponding preimage of 1, we see that there is a bijection between 2 and P (S), where each function is the characteristic function of the subset in P (S) with which it is identified . Hence 2 and P (S) could be considered identical set - theoretically . (Thus there are two distinct notational motivations for denoting the power set by 2: the fact that this function - representation of subsets makes it a special case of the X notation and the property, mentioned above, that 2 = 2 .) </P>

The cardinality of the power set of 0 1 2 ... 10 is