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:

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

More precisely:

Let $F$ be the mapping on the universal class defined by letting:

$F \left({a}\right) = \bigcup a$

for each set $a$.

Let $G$ be the mapping on the natural numbers defined recursively by letting:

$G \left({0}\right) = \left\{ {x}\right\}$

$G \left({n^+}\right) = F \left({G \left({n}\right)}\right)$

for each natural number $n$.

Then the transitive closure of $x$ is defined as the union of the image of $G$.

Sources

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.): $\S 15.1$