criterion for a set to be transitive


A set X is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath if and only if its power setMathworldPlanetmath P(X) is transitive.


First assume X is transitive. Let AB𝒫(X). Since B𝒫(X), BX. Thus, AX. Since X is transitive, AX. Hence, A𝒫(X). It follows that 𝒫(X) is transitive.

Conversely, assume 𝒫(X) is transitive. Let aX. Then {a}𝒫(X). Since 𝒫(X) is transitive, {a}𝒫(X). Thus, a𝒫(X). Hence, aX. 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