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Math Help - Cartesian Product in set theory

  1. #1
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    Cartesian Product in set theory

    Hi Folks,

    I have came across a need and I am not aware if there is such a thing in set theory and if so what is it called.

    Mainly I have several sets that I am interested in their cartesian product. But this cartesian product should not be a set of ordered pairs but a set of sets. Basically unordered pairs.

    I wonder if this concept is well defined and what is it called.

    Thanks.
    P.
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  2. #2
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    You mean something like:

    <br />
\{ \{x_1,x_2,\ldots\} \mid x_i\in X_i\}<br />

    I can't think of any neat way to denote this. There's of course the idea of unordered n-tuples from a set A, which is denoted

    <br />
[A]^n<br />

    But that's different than what you stated as it's n-tuples from 1 set, and not from a collection.
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  3. #3
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    The notation you have used is exactly what I need. But will it be true to call it as Cartesian Product?
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  4. #4
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    No. Cartesian products are ordered. If you have say

    <br />
[A]^n<br />

    This is also not a cartesian product. Cartesian products are an ordered n-tuple. The family of unordered n-tuples I don't think has any interesting name, except the set of unordered n-tuples! I never ran accross a reason to have to talk about them the way you did, as from a family of sets. The set of unordered n-tuples from a set is useful many areas, mostly when you talk about colorings (because normally you want to color say pairs of vertices (ie. an edge) rather than a binary relation on vertices (ie. an arc))
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  5. #5
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    Indeed not the unordered tuples of a single set but many of them is needed.

    For the coloring problem I think what we need is the set of all sets of size 2, which is a memebr of the powerset of the set of vertexes.
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