# criterion for a set to be transitive

###### Theorem.

A set $X$ is transitive^{} if and only if its power set^{} $\mathrm{P}\mathit{}\mathrm{(}X\mathrm{)}$ is transitive.

###### Proof.

First assume $X$ is transitive. Let $A\in B\in \mathcal{P}(X)$. Since $B\in \mathcal{P}(X)$, $B\subseteq X$. Thus, $A\in X$. Since $X$ is transitive, $A\subseteq X$. Hence, $A\in \mathcal{P}(X)$. It follows that $\mathcal{P}(X)$ is transitive.

Conversely, assume $\mathcal{P}(X)$ is transitive. Let $a\in X$. Then $\{a\}\in \mathcal{P}(X)$. Since $\mathcal{P}(X)$ is transitive, $\{a\}\subseteq \mathcal{P}(X)$. Thus, $a\in \mathcal{P}(X)$. Hence, $a\subseteq X$. It follows that $X$ is transitive. ∎

Title | criterion for a set to be transitive |
---|---|

Canonical name | CriterionForASetToBeTransitive |

Date of creation | 2013-03-22 16:18:23 |

Last modified on | 2013-03-22 16:18:23 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 6 |

Author | Wkbj79 (1863) |

Entry type | Theorem |

Classification | msc 03E20 |

Related topic | CumulativeHierarchy |