not) always has strictly higher cardinality than the set itself (or 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 , 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 forms an abelian group when it is 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. Representing subsets as functions In set theory, is the notation representing the set of all functions from to . As "2" can be defined as (see, for example, von Neumann ordinals), (i.e., ) is the set of all functions from to . As shown above, and the power set of , , is considered identical set-theoretically. This equivalence can be applied to the example above, in which , to get the isomorphism with the binary representations of numbers from 0 to , with being the number of elements in the set or . First, the enumerated set is defined in which the number in each ordered pair represents the position of the paired element of in a sequence of binary digits such as ; of is located at the first from the right of this sequence and is at the second from the right, and 1 in the sequence means the element of corresponding to the position of it in the sequence exists in the subset of for the sequence while 0 means it does not. For the whole power set of