Correct notation for union of all elements in a set?

Question

Say I have a set $H$ and I want to describe the union of all elements in $H$. How would I write that? I believe I've seen a big U with a subscript used before.

Answer 1

If the elements of $H$ are sets, it would make sense to write $$\bigcup_{h\in H} h$$ However, if the elements of $H$ are not sets, taking the union of them would not make sense. In this case, the totality of all elements of $H$ is just $H$.

Answer 2

In set theory we use $\bigcup H$ to denote the union of all the elements of $H$. Sometimes we write it explicitly, in one of several ways:

1. $\bigcup_{h\in H}h$, or $\bigcup\limits_{h\in H}h$,
2. $\bigcup\{h\mid h\in H\}$ (this is useful when $H$ is not assigned a variable, but defined via a formula),
3. $\{x\mid\exists h\in H:x\in h\}$,
4. $\bigcup H$, as I remarked before.

Note, however, that in set theory it is often the case that everything is a set, so taking the union makes sense.

If you are not working in set theory, then I would (1) recommend using the first notation, and (2) be sure that the elements of $H$ are sets, or that you defined a notion similar to union over those objects. Taking the union of two points on the plane doesn't make much sense if you don't consider them as sets.