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 sethas n elements then the power sethas
elements. In general we write the power set of a setas
The setabove has 3 elements so the power sethas
elements and the set
has 2 elements so the power set
has
elements.
Any subset of the power set of a setis called a family of sets over
The cardinality of the power set of a setis 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.