PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] criterion for a set to be transitive (Theorem)
Theorem   A set $ X$ is transitive if and only if its power set % latex2html id marker 165 $ \mathcal{P}(X)$ is transitive.
Proof. First assume $ X$ is transitive. Let % latex2html id marker 172 $ A \in B \in \mathcal{P}(X)$. Since % latex2html id marker 174 $ B \in \mathcal{P}(X)$, $ B \subseteq X$. Thus, $ A \in X$. Since $ X$ is transitive, $ A \subseteq X$. Hence, % latex2html id marker 184 $ A \in \mathcal{P}(X)$. It follows that % latex2html id marker 186 $ \mathcal{P}(X)$ is transitive.

Conversely, assume % latex2html id marker 188 $ \mathcal{P}(X)$ is transitive. Let $ a \in X$. Then % latex2html id marker 192 $ \{a\} \in \mathcal{P}(X)$. Since % latex2html id marker 194 $ \mathcal{P}(X)$ is transitive, % latex2html id marker 196 $ \{a\} \subseteq \mathcal{P}(X)$. Thus, % latex2html id marker 198 $ a \in \mathcal{P}(X)$. Hence, $ a \subseteq X$. It follows that $ X$ is transitive. $ \qedsymbol$



"criterion for a set to be transitive" is owned by Wkbj79.
(view preamble)

View style:

See Also: cumulative hierarchy


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: power set, transitive
There is 1 reference to this entry.

This is version 3 of criterion for a set to be transitive, born on 2006-10-08, modified 2006-11-06.
Object id is 8429, canonical name is CriterionForASetToBeTransitive.
Accessed 859 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy
gap in my education by pahio on 2006-10-08 02:27:33
I know no transitive sets. Please give a concrete example!
[ reply | up ]
Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)