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A set is transitive if and only if all of its elements are subsets of it.

Equivalently, a set $A$ is transitive if and only if:

  • it contains its union
  • the powerset of $A$ contains $A$
  • all members of the members of $A$ are members of $A$

Properties of Transitive Sets

If $A$ is transitive, then if $x$ and $A$ are connected somehow by membership (that is, $x \in y \in z \ldots \in A$), then $x \in A$.

The intersection of two transitive sets is transitive.

In set theory, transitive sets play an important role in models of ZFC. See transitive ZFC model.