Definition:Set Union/Set of Sets

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Let $\mathbb S$ be a set of sets.

The union of $\mathbb S$ is:

$\displaystyle \bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$

That is, the set of all elements of all elements of $\mathbb S$.

Remark: The existence of $\displaystyle \bigcup \mathbb S$ is an independent axiom.

Thus the general union of two sets can be defined as:

$\displaystyle \bigcup \set {S, T} = S \cup T$

Also denoted as

Some sources denote $\displaystyle \bigcup \mathbb S$ as $\displaystyle \bigcup_{S \mathop \in \mathbb S} S$.


Set of Arbitrary Sets


\(\displaystyle A\) \(=\) \(\displaystyle \set {1, 2, 3, 4}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {a, 3, 4}\) $\quad$ $\quad$
\(\displaystyle C\) \(=\) \(\displaystyle \set {2, a}\) $\quad$ $\quad$

Let $\mathscr S = \set {A, B, C}$.


$\displaystyle \bigcup \mathscr S = \set {1, 2, 3, 4, a}$

Set of Initial Segments

Let $\Z$ denote the set of integers.

Let $\map \Z n$ denote the initial segment of $\Z_{> 0}$:

$\map \Z n = \set {1, 2, \ldots, n}$

Let $\mathscr S := \set {\map \Z n: n \in \Z_{> 0} }$

That is, $\mathscr S$ is the set of all initial segments of $\Z_{> 0}$.


$\displaystyle \bigcup \mathscr S = \Z_{> 0}$

that is, the set of strictly positive integers.

Also see