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Math Help - Cross Product

  1. #1
    Newbie nickgc's Avatar
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    Cross Product

    For finite sets M and N and P, if the cross product of M and N is equal to the cross product of M and P, then N = P. (if MxN=MxP then N=P)

    my prof told us its FALSE. but why?
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  2. #2
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    Re: Cross Product

    This is also false for numerical product for a similar reason.
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  3. #3
    Newbie nickgc's Avatar
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    Re: Cross Product

    what's the reason?
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  4. #4
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    Re: Cross Product

    First say if you think the following fact is true for numbers:

    For all m, n, p, if mn = mp, then n = p.

    If you think this is true, try proving it. Pay special attention so that all operations you do in the proof are well-defined.
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  5. #5
    Newbie nickgc's Avatar
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    Re: Cross Product

    Now i get it:

    For all m, n, p, if mn = mp, then n = p:

    for example

    if m = 0, n = 2, p = 100

    then: mn = 0, mp = 0 and mn = mp
    but n is not equal to p

    am i right?
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  6. #6
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    Re: Cross Product

    By the way, when A and B are sets, A\times B is the "Cartesian product", not the "cross product".
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  7. #7
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    Re: Cross Product

    Yes.

    Now consider some sets M, N, P with 0, 2, and 100 elements, respectively.
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  8. #8
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    Re: Cross Product

    Now consider some sets M, N, P with 0, 2, and 100 elements, respectively.
    M = {0}
    N = {2}
    P = {100}

    MxN = {(0,2)}
    MxP = {(0,100)}

    how would MxN be equal to MxP?
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  9. #9
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    Re: Cross Product

    Quote Originally Posted by nickgc View Post
    For finite sets M and N and P, if the cross product of M and N is equal to the cross product of M and P, then N = P. (if MxN=MxP then N=P) my prof told us its FALSE. but why?
    Consider: \emptyset  = \emptyset  \times \{ 1\}  = \emptyset  \times \{ 2\}
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  10. #10
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    Re: Cross Product

    in ordinary language, if we never pick a first coordinate, it doesn't matter which second coordinate we were going to pick, it never gets chosen.

    the same logic goes for the second coordinate: no matter which first coordinate we pick, if we never choose the second coordinate, we never get a pair.

    picking an element of the empty set pretty much stops us dead in our tracks, it has no elements to pick.
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  11. #11
    Newbie nickgc's Avatar
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    Re: Cross Product

    Consider: (empty set) = (empty set) x {1} = (empty set) x {2}
    OK now i understand it.. thank you very much
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