# Definition:Transitive Closure (Set Theory)

## Definition

### Definition 1

Let $x$ be a set.

Then the **transitive closure** of $x$ is the smallest transitive superset of $x$.

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

### Definition 2

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:

- $\left\{ {x}\right\}, x, \bigcup x, \bigcup^2 x, \dots, \bigcup^n x, \dots$

## Notation

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

## Also see

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 9.2$