If S is the set {A, B, C} then {A,C} is a subset of S. There are other subsets of S; the complete list is as follows:
So the power set of S, written P(S), is the set containing all the subsets above. Written out this would be the set:
P(S) = { {}, {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} }
If n = S is the number of elements of S, then the respective power set contains P(S) = 2^{n} elements. (One can [and computers actually do] represent the elements of P(S) as nbit numbers; the nth bit refers to presence or absence of the nth element of S. There are 2^{n} such numbers.)
One can also consider the power set of infinite sets. Cantor's diagonal argument shows that the power set of an infinite set always has strictly higher cardinality than the set itself (informally the power set must be 'more infinite' than the original set). The power set of the natural numbers for instance can be put in a onetoone correspondence with the set of real numbers (by identifying an infinite 01 sequence with the set of indices where the ones occur).
The power set of a set S, together with the operations of union, intersection and complement forms 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 S. For infinite boolean algebras this is no longer true, but every infinite boolean algebra is a subalgebra of a power set boolean algebra.
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