Definition:Transitive Closure of Set

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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:

$\set x, x, \bigcup x, \bigcup^2 x, \dots, \bigcup^n x, \dots$

Notation

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

Also see

• Transitive Closure of Set Always Exists

• Equivalence of Definitions of Transitive Closure of Set

Sources

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