axiom of power set
The axiom of power set is an axiom of Zermelo-Fraenkel set theory which postulates that for any set there exists a set , called the power set of , consisting of all subsets of . In symbols, it reads:
In the above, is defined as . By the extensionality axiom, the set is unique.
The Power Set Axiom allows us to define the Cartesian product of two sets and :
The Cartesian product is a set since
We may define the Cartesian product of any finite collection of sets recursively:
Title | axiom of power set |
---|---|
Canonical name | AxiomOfPowerSet |
Date of creation | 2013-03-22 13:43:03 |
Last modified on | 2013-03-22 13:43:03 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 11 |
Author | mathcam (2727) |
Entry type | Axiom |
Classification | msc 03E30 |
Synonym | power set axiom |
Synonym | powerset axiom |
Synonym | axiom of powerset |