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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.

Git Gud
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EmutheEmu
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2 Answers2

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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.

Asaf Karagila
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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 $H$.

paw88789
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  • The elements of the set are volumes and I'm trying to describe the union of the volumes. Does this still make sense? – EmutheEmu Oct 10 '14 at 23:39
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    I suppose it might if you're thinking of each volume as a set of points in say $3$-space. Although I think technically the volume is a measure of the set of points, not the set itself. – paw88789 Oct 10 '14 at 23:41