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  1. #1
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    Sets

    How can I prove/disprove that the product of two uncountable sets can be countable?

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  2. #2
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by Nickolase View Post
    How can I prove/disprove that the product of two uncountable sets can be countable?

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    One way is if their product is the set of integers such as x as the set of positive real numbers multiplied by 1/x which is the set of 1's. Intergers are countable. That is one way.
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  3. #3
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Nickolase View Post
    How can I prove/disprove that the product of two uncountable sets can be countable?

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    Quote Originally Posted by oldguynewstudent View Post
    One way is if their product is the set of integers such as x as the set of positive real numbers multiplied by 1/x which is the set of 1's. Intergers are countable. That is one way.
    I don't have an answer, just a comment for clarity. Does the original question refer to the Cartesian product of the two sets, or something different?
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  4. #4
    Junior Member guildmage's Avatar
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    Start by trying to show whether there is (or is not) a one-to-one correspondence between the cross-product of the two sets and the set of natural numbers.
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  5. #5
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    1. prove NxN is equinumerous to N
    2. for uncountable, prove a set (any one of the two sets) is equinumerous to a proper subset of the product of the two sets. thus if product is countable we run into a contradiction
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  6. #6
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    First, if B injects into C and B is uncountable, then C is uncountable.
    Proof (here 'w', read 'omega', stands for the set of natural numbers):
    Toward a contradiction, suppose B injects into C, and B is uncountable, but C is countable.
    So C injects into w.
    So, since B injects in C, we have B injects into w, so B is countable. Done.

    Suppose S and T are uncountable.

    Let y be in T.

    S X {y} is equinmerous with S.

    So S X {y} is uncountable.

    S X {y} is a subset of S X T.

    So S X T is uncountable.
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