# Denumerability

• Nov 14th 2010, 11:52 AM
veronicak5678
Denumerability
Give an example, if possible, of a denumerable collection of finite sets whose union is finite.
• Nov 14th 2010, 12:00 PM
FernandoRevilla
Quote:

Originally Posted by veronicak5678
Give an example, if possible, of a denumerable collection of finite sets whose union is finite.

Choose

$A_n=\left\{{a}\right\}\;(n\in{\mathbb{N}})$

Regards.
• Nov 14th 2010, 12:03 PM
veronicak5678
Thanks for answering, but I don't understand the set. What is {a}?
• Nov 14th 2010, 12:06 PM
FernandoRevilla
Quote:

Originally Posted by veronicak5678
Thanks for answering, but I don't understand the set. What is {a}?

For example, $a=1$.

Regards.
• Nov 14th 2010, 12:07 PM
Jhevon
Quote:

Originally Posted by veronicak5678
Thanks for answering, but I don't understand the set. What is {a}?

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, 12:11 PM
veronicak5678
Oh! I see. I didn't understand where the n came into it. Thanks.
• Nov 14th 2010, 01:39 PM
MoeBlee
Quote:

Originally Posted by veronicak5678
Give an example, if possible, of a denumerable collection of finite sets whose union is finite.

What is the exact definition of 'denumerable' here? If it is "x is denumerable <-> x is 1-1 with the set of natural numbers", then there is no example of a denumerable set whose union is finite, and that is easy to prove.
• Nov 14th 2010, 02:42 PM
FernandoRevilla
Quote:

Originally Posted by MoeBlee
What is the exact definition of 'denumerable' here? If it is "x is denumerable <-> x is 1-1 with the set of natural numbers", then there is no example of a denumerable set whose union is finite, and that is easy to prove.

I understand and you are right with that interpretation. Another interpretation:

If we say

"Let $\left\{{A_n:\;n\in{\mathbb{N}}}\right\}$ a denumerable family of indexed sets"

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

And my example is related to this interpretation.

Regards.
• Nov 14th 2010, 03:01 PM
Jhevon
Quote:

Originally Posted by MoeBlee
What is the exact definition of 'denumerable' here? If it is "x is denumerable <-> x is 1-1 with the set of natural numbers", then there is no example of a denumerable set whose union is finite, and that is easy to prove.

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, 03:05 PM
Plato
Quote:

Originally Posted by Jhevon
the sets were not required to be denumerable, the collection of sets are to be. each set should be finite. i agree with FernandoRevilla's interpretation. And i don't think it would work without making a lot of the sets (a countably infinite amount of them) the same set.

But I do not consider that collection to be denumerable.
That collection has only one member: {a}.
• Nov 14th 2010, 03:14 PM
Jhevon
Quote:

Originally Posted by Plato
But I do not consider that collection to be denumerable.
That collection has only one member: {a}.

well, yes. haha, i had the hope we could get away with thinking of the collection as: $\displaystyle \{ A_1, A_2, A_3, \dots \}$ as opposed to $\displaystyle \{ \{ a \}, \{ a \}, \{ a \}, \dots \} = \{ \{ a \} \}$ 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, 05:52 PM
topspin1617
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, 06:22 PM
Drexel28
Quote:

Originally Posted by topspin1617
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 :)

Well, clearly the issue here is whether or not given an indexed family of sets $f:\mathcal{A}\to\text{Universe}:\alpha\to U_{\alpha}$ whether we're considering the function itself as feasible $f:\mathbb{N}\to\mathbb{R}:n\mapsto A_n=\{0\}$ or if we must consider the image $f\left(\mathcal{A}\right)$
• Nov 15th 2010, 10:24 AM
MoeBlee
Quote:

Originally Posted by Jhevon
the sets were not required to be denumerable, the collection, or "number", of sets are to be.

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.