Power set

In mathematics, given a set S, the power set of S, written P(S) or 2^S, is the set of all subsets of S. In formal language, the existence of power set of any set is presupposed by the axiom of power set.

In this case S is usually called the universal set and any subset F of P(S) is called a family of sets over S.

For example, if S is the set {A, B, C} then the complete list of subsets of S is as follows:

• {} (the empty set)
• {A}
• {B}
• {C}
• {A, B}
• {A, C}
• {B, C}
• {A, B, C}

and hence the power set of S is

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)| = 2ⁿ elements. (One can - and computers actually do - represent the elements of P(S) as n-bit numbers; the n-th bit refers to presence or absence of the n-th element of S. There are 2ⁿ such numbers.)

One can also consider the power set of infinite sets. Cantor's diagonal argument shows that the power set of a set (infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be 'greater' than the original set). The power set of the natural numbers for instance can be put in a one-to-one correspondence with the set of real numbers (by identifying an infinite 0-1 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. For infinite boolean algebras this is no longer true, but every infinite boolean algebra is a subalgebra of a power set boolean algebra.

The notation 2^S

In set theory, X^Y is the set of all functions from Y to X. As 2 can be defined as {0,1} (see natural number), 2^S is the set of all functions from S to {0,1}. By identifying a function in 2^S with the corresponding preimage of 0, we see that there is a bijection between 2^S and P(S), hence 2^S and P(S) could be considered identical set-theoretically.