Definition 1. A set $T$ is transitive if every element of $T$ is a subset of $T$.
Proposition 1. The following statements are equivalent:
-
$T$ is transitive.
- \[\bigcup T \subseteq T.\]
- \[T \subseteq \powerset(T).\]
References
- [1]
- Wikipedia contributors. Transitive set. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 13.