Give an example, if possible, of a denumerable collection of finite sets whose union is finite.

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- Nov 14th 2010, 10:52 AMveronicak5678Denumerability
Give an example, if possible, of a denumerable collection of finite sets whose union is finite.

- Nov 14th 2010, 11:00 AMFernandoRevilla
- Nov 14th 2010, 11:03 AMveronicak5678
Thanks for answering, but I don't understand the set. What is {a}?

- Nov 14th 2010, 11:06 AMFernandoRevilla
- Nov 14th 2010, 11:07 AMJhevon
haha, it's a letter, well, a set containing a letter. doesn't matter what a is. all your sets have one element, and that is called a. you have a denumerable collection of 1-element sets (hence finite sets), whose union is a 1-element set (hence finite). if it bothers you so much, you can replaced a with whatever you want. like a number or something.

- Nov 14th 2010, 11:11 AMveronicak5678
Oh! I see. I didn't understand where the n came into it. Thanks.

- Nov 14th 2010, 12:39 PMMoeBlee
- Nov 14th 2010, 01:42 PMFernandoRevilla
I understand and you are right with that interpretation. Another interpretation:

If we say

"Let $\displaystyle \left\{{A_n:\;n\in{\mathbb{N}}}\right\}$ a denumerable family of__indexed__sets"

there is no problem if we consider $\displaystyle A_p=A_q$ for some $\displaystyle p\neq q$.

And my example is related to this interpretation.

Regards. - Nov 14th 2010, 02:01 PMJhevon
the sets were not required to be denumerable, the collection, or "number", of sets are to be. each set should be finite. i agree with

**FernandoRevilla**'s interpretation (he wasn't very rigorous in his definition to begin with, but i think he got the idea across). And i don't think it would work without making a lot of the sets (a countably infinite amount of them) the same set. - Nov 14th 2010, 02:05 PMPlato
- Nov 14th 2010, 02:14 PMJhevon
well, yes. haha, i had the hope we could get away with thinking of the collection as: $\displaystyle \displaystyle \{ A_1, A_2, A_3, \dots \}$ as opposed to $\displaystyle \displaystyle \{ \{ a \}, \{ a \}, \{ a \}, \dots \} = \{ \{ a \} \}$ but if we can't (and i can see how it would make sense that we can't) then it would be impossible to give such an example.

- Nov 14th 2010, 04:52 PMtopspin1617
Maybe they are distinguishing between a "collection of sets" and a "set of sets" to allow something like the solution given here.... maybe a "collection" means a multiset? =P

Ah, ambiguities :) - Nov 14th 2010, 05:22 PMDrexel28
Well, clearly the issue here is whether or not given an indexed family of sets $\displaystyle f:\mathcal{A}\to\text{Universe}:\alpha\to U_{\alpha}$ whether we're considering the function itself as feasible $\displaystyle f:\mathbb{N}\to\mathbb{R}:n\mapsto A_n=\{0\}$ or if we must consider the image $\displaystyle f\left(\mathcal{A}\right)$

- Nov 15th 2010, 09:24 AMMoeBlee
If you tell me "X is a denumerable collection of finite sets whose union is finite" then I take that as:

X is denumerable

Every member of X is finite

The union of X is finite

The word "collection" has no special meaning here other than 'set' unless otherwise specified.

In particular, it's a jump to regard "collection" as an indexing function unless that is specified.