Power sets allow us to discuss the class of all subsets of a given set , i.e. . That this is a set is the subject of the Power Set Axiom.
Given a set there exists a set of sets such that iff .
Theorem Given a set , there exists a unique set whose elements are the subsets of .
Proof If and are two such sets of subsets then if and only if . But the same is true of . Thus iff , and so by the Axiom of Extensionality.
Definition Given a set , the set of all subsets of is called the power set of . It is denoted .
Example If then .
Recall the Kuratowski definition of an ordered pair, for and elements of a set . Note that and are both subsets of , i.e. they are elements of the power set .
This means that is a subset of , i.e. .
We can generalise this slightly with a simple trick. We can define with and for sets and . In order to do this, we simply take the elements and from the union of sets .
In other words, we have with and .
Theorem The class of all ordered pairs of elements of with and , is a set.
Proof The set in question is given by . This is a set by the axioms of Power Set, Union and the Axiom Schema of Comprehension.
Definition The set of ordered pairs with and is called the cartesian product of and , and is denoted .