A set $A$ is **transitive** if it satisfies any of the following equivalent conditions:

- each element of an element of $A$ is an element of $A$
- each element of $A$ is a subset of $A$
- $\bigcup A\subseteq A$
- $A \subseteq \mathcal{P}(A)$

The word *transitive* is used because $a\in b\in A$ implying $a\in A$ reminds the transitivity in order relations. In fact, for ordinals, set membership will be an order relation