The Power Set
The power setofis the set of all subsets ofincluding the empty setand itself.
Properties of the Power Set
If the sethas n elements then the power sethaselements. In general we write the power set of a setas The setabove has 3 elements so the power sethaselements and the sethas 2 elements so the power sethaselements.
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.