## The Power Set

The power set of is the set of all subsets of including the empty set and itself.

Example: If then Example: If then Properties of the Power Set

If the set has n elements then the power set has elements. In general we write the power set of a set as The set above has 3 elements so the power set has elements and the set has 2 elements so the power set has elements.

Any subset of the power set of a set is called a family of sets over The cardinality of the power set of a set is strictly larger than the cardinality of  The power set of a set A with the operations of union, intersection and complement can be viewed as an example of a Boolean algebra. We 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 not true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra.

The power set of a set A 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 be shown that the power set considered together with both of these operations forms a Boolean ring. 