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

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:

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

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

• I would write $$\mathcal G:=\left\{\bigcup_{a\in A}F_a:A\subseteq\{1,2,\ldots,n\}\right\}$$
– CIJ
Commented Sep 24, 2015 at 20:02

The simplest solution is

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

• I would have opted for \cal and not \scr, though. :-) Commented Sep 24, 2015 at 20:02
• @Asaf: Most do, I think, but I find \cal ugly, and \scr is mostly fairly close to my handwritten symbols. Commented Sep 24, 2015 at 20:03
• I like both, depending on the context. In my recent work I have both $\cal G,F$ and $\scr G,F$'s, and it's a good thing my handwriting allows discerning them! ;) Commented Sep 24, 2015 at 20:05
• @BrianM.Scott this looks good, although since $\mathscr{A}$ is a set in the collection of sets $\mathscr{F}$ I am wondering if we can write $$\mathscr{G}=\left\{\bigcup\mathscr{A}:\mathscr{A}\in \mathscr{F}\right\}\;.$$? Or is this incorrect? Commented Sep 24, 2015 at 21:35
• @möbius: That would be incorrect. $\mathscr{A}$ is not a set in the collection $\mathscr{F}$: it's a subcollection of $\mathscr{F}$. Taking the union of every possible subcollection of $\mathscr{F}$ gives you every possible union of members or $\mathscr{F}$. Commented Sep 25, 2015 at 1:22

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: $$\mathcal G=\left\{\bigcup_{i\in I} F_i\mathrel{}\middle|\mathrel{} I\subseteq\{1,\ldots,n\}\right\}$$