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Power set

The power set,  or powerset of a given , written or , is the set of all subsets of .

In the ZFC axiom system , the existence of the power set of any set is postulated by the Powerset Axiom.

Any subset F of is called a family of sets over .

For instance, if is a set of three elements then the complete list of subsets of is as follows:

, the empty set

are the subsets with one element

are the subsets with two elements

is the only subset with three elements

The power set is closely related to the binomial theorem. The number of sets with   elements in the power set of a set   with elements is given by a binomial coefficient , also denoted by .  coefficient. For example the power set of a set with three elements, has:

subset with no element

subsets with 1 element

subsets with 2 elements

subset with 3 elements

Thus  .

To see more examples visit

Note that if is then .

The above mentioned notation has its root in the following facts:

We can identify every subset of a set with its image under the indicator function . Then

If is finite , then the cardinality . More generally, the so-called Cantor's diagonal argument shows that the power set of any set (finite or not) always has strictly higher cardinality than the set itself.

In the set theory the notation is used for the set of all functions from to . Since the set appears in some axiomatic definitions of non-negative integers as a representation for , the notation is at hand.  

The power set of a set endowed with the operations of union, intersection and complement is the prototypical example of a Boolean algebra . One can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. Note that for infinite Boolean algebras this statement is no longer true. On the other hand,  every infinite Boolean algebra is a subalgebra of a power set Boolean algebra.

Another type of examples we get starting with the observation that the power set of a set forms an Abelian group with respect to the operations of symmetric difference. Here the empty set plays the role of the  unit and each set being its own inverse. Similarly, is a commutative semigroup when considered with the operation of intersection. Since intersection is distributive with respect to symmetric difference the power set equipped with these two operations forms a commutative ring .

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