criterion for a set to be transitive

Theorem.

A set $X$ is transitive if and only if its power set $\mathcal{P}(X)$ 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