Definition:Set Union/Finite Union

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

Then:
 * $\displaystyle S = \bigcup_{i \mathop \in \N^*_n} S_i = \left\{{x: \exists i \in \N^*_n: x \in S_i}\right\}$

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

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

Also defined as
The specific nature of the indexing set $\N^*_n$ is immaterial; some treatments may use a zero-based set, thus:
 * $\displaystyle S = \bigcup_{i \mathop \in \N_n} S_i = \left\{{x: \exists i \in \N_n: x \in S_i}\right\}$

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

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:
 * $\displaystyle \bigcup_{i \mathop = 1}^n S_i$
 * $\displaystyle \bigcup_{1 \mathop \le i \mathop \le n} S_i$

Also see

 * Finite Union is Union of Indexed Family of Sets