Give an example, if possible, of a denumerable collection of finite sets whose union is finite.
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.
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.
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.
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.
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 $\displaystyle f\left(\mathcal{A}\right)$
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.