# axiom of power set

The *axiom of power set ^{}* is an axiom of Zermelo-Fraenkel set theory

^{}which postulates

^{}that for any set $X$ there exists a set $\mathcal{P}(X)$, called the

*power set*of $X$, consisting of all subsets of $X$. In symbols, it reads:

^{}$$\forall X\exists \mathcal{P}(X)\forall u(u\in \mathcal{P}(X)\leftrightarrow u\subseteq X).$$ |

In the above, $u\subseteq X$ is defined as $\forall z(z\in u\to z\in X)$. By the extensionality axiom, the set $\mathcal{P}(X)$ is unique.

The Power Set Axiom allows us to define the Cartesian product^{} of two sets $X$ and $Y$:

$$X\times Y=\{(x,y):x\in X\wedge y\in Y\}.$$ |

The Cartesian product is a set since

$$X\times Y\subseteq \mathcal{P}(\mathcal{P}(X\cup Y)).$$ |

We may define the Cartesian product of any finite collection^{} of sets recursively:

$${X}_{1}\times \mathrm{\cdots}\times {X}_{n}=({X}_{1}\times \mathrm{\cdots}\times {X}_{n-1})\times {X}_{n}.$$ |

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 |