axiom of power set

The axiom of power setMathworldPlanetmath is an axiom of Zermelo-Fraenkel set theoryMathworldPlanetmath which postulatesMathworldPlanetmath that for any set X there exists a set 𝒫(X), called the power setMathworldPlanetmath of X, consisting of all subsets of X. In symbols, it reads:


In the above, uX is defined as z(zuzX). By the extensionality axiom, the set 𝒫(X) is unique.

The Power Set Axiom allows us to define the Cartesian productMathworldPlanetmath of two sets X and Y:


The Cartesian product is a set since


We may define the Cartesian product of any finite collectionMathworldPlanetmath 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