Set notation question: Write collection that contains all possible unions of sets from another collection

Question

Suppose $\mathcal{F}$ is a finite collection of sets $F_{1}, F_{2}, F_{3} \ldots, F_{n}$.

I want to construct a collection of sets $\mathcal{G}$ that contains every possible union of sets in $\mathcal{F}$. For example, I want $\mathcal{G}$ to contain $F_{1}\cup F_{1}$, $F_{1}\cup F_{2} \cup F_{3} \cup F_{4}$, $\bigcup_{j=1}^{n}F_{j}$, etc. etc.

However, I am not sure how to write $\mathcal{G}$ using set notation.

My guess was something like:

\begin{equation} \mathcal{G} =:\left\{\bigcup_{i\geq1} G_{i}: \forall_{i} \text{ }G_{i} \in \mathcal{F} \text{ }\right\} \end{equation}

Is this correct? If not can someone provide the correct notation?

Answer 1

The simplest solution is

$$\mathscr{G}=\left\{\bigcup\mathscr{A}:\mathscr{A}\subseteq\mathscr{F}\right\}\;.$$

Answer 2

If I had seen the notation you suggest, I would have been quite confused about it. If there is a clear explanation in text, it might have helped, but unfortunately not everyone provides textual explanation, and it's not always clear.

I'd prefer to use the following: $$\mathcal G=\left\{\bigcup_{i\in I} F_i\mathrel{}\middle|\mathrel{} I\subseteq\{1,\ldots,n\}\right\}$$