Definition:Transitive Closure (Set Theory)

Definition
The transitive closure of a set $S$ is the smallest set $T$ that contains $S$ and is transitive.

Definition 2
The following is not equivalent to the above, but they are almost the same.

Notation
The transitive closure may be denoted $\operatorname{TrCl}$.

Also see

 * Transitive Closure Always Exists (Set Theory)