## Definition A set is transitive if and only if all of its elements are subsets. 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 <a href="Transitive_ZFC_model" class="mw-redirect" title="Transitive ZFC model">transitive ZFC model</a>.