Definition – Set union. – Math-reference.net

Definition

Let $A, B$ be sets. The union of two sets, written $A\cup{B}$ , is defined as,

$A\cup{B}$ = $\{x~|~x\in{A}~\vee~x\in{B}\}$

We can extend this idea to the union of three or more sets. Given a family of sets $\mathcal{F}$ , we write the union of all sets in $\mathcal{F}$ as

$\bigcup\limits_{i} \mathcal{F}_{i}$

As long as $\mathcal{F}$ has a finite number of sets. If, however, the family of sets is either infinite, or we are unable to assign an index to each of the sets within $\mathcal{F}$ , then we can still write the union of all the sets in $\mathcal{F}$ as follows,

$\bigcup\limits_{B\in \mathcal{F}} B_{i} = \{x~|~\exists{B\in{\mathcal{F}}}~s.t~x\in{B}\}$

Notes

Ie, if there exists a set $B\in{F}$ , then we take its elements and add it to our union, doing so for all $B$ in $F$ .