Definition:Set Union/Finite Union

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Definition

Let $S = S_1 \cup S_2 \cup \ldots \cup S_n$.

Then:

$\ds S = \bigcup_{i \mathop \in \N^*_n} S_i = \set {x: \exists i \in \N^*_n: x \in S_i}$

where $\N^*_n = \set {1, 2, 3, \ldots, n}$.


If it is clear from the context that $i \in \N^*_n$, we can also write $\ds \bigcup_{\N^*_n} S_i$.



This article, or a section of it, needs explaining.
In particular: We need a proof that $S_1 \cup S_2 \cup \cdots \cup S_n$ equals the indexed union. See talk page. I've thought about it, Dfeuer's right.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.



Also defined as

The specific nature of the indexing set $\N^*_n$ is immaterial; some treatments may use a zero-based set, thus:

$\ds S = \bigcup_{i \mathop \in \N_n} S_i = \set {x: \exists i \in \N_n: x \in S_i}$

where $\N_n = \set {0, 1, 2, \ldots, n - 1}$.

In this context the sets under consideration are $S = S_0 \cup S_1 \cup \ldots \cup S_{n - 1}$.

The distinction is sufficiently trivial as to be hardly worth mentioning.


Also denoted as

Other notations for this concept are:

$\ds \bigcup_{i \mathop = 1}^n S_i$
$\ds \bigcup_{1 \mathop \le i \mathop \le n} S_i$


Sources

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