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Math Help - Denumerability

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    Denumerability

    Give an example, if possible, of a denumerable collection of finite sets whose union is finite.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    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.
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    Thanks for answering, but I don't understand the set. What is {a}?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    Thanks for answering, but I don't understand the set. What is {a}?

    For example, a=1.

    Regards.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by veronicak5678 View Post
    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.
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    Oh! I see. I didn't understand where the n came into it. Thanks.
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    Quote Originally Posted by veronicak5678 View Post
    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.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by MoeBlee View Post
    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.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by MoeBlee View Post
    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.
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    Quote Originally Posted by Jhevon View Post
    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}.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    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.
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    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
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by topspin1617 View Post
    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)
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    Quote Originally Posted by Jhevon View Post
    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.
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