Definition:Transitive Closure (Set Theory)/Definition 2

Definition
Let $x$ be a set.

For each natural number $n \in \N_{\ge 0}$ let:


 * $\bigcup^n x = \underbrace{\bigcup \bigcup \cdots \bigcup}_{n} x$

Then the transitive closure of $x$ is the union of the sets $\{x\}, x, \bigcup x, \bigcup^2 x, \dots, \bigcup^n x, \dots$.