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Math Help - relations

  1. #1
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    relations

    Let B={1,2,3.....m), let R be a relation on B and its antisymmetric

    a) What is the biggest number of ordered pairs that can be in R?

    b)How many antisymmetric relations on A have the size found in a) ??
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  2. #2
    Senior Member Shanks's Avatar
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    (a)Since the matrix representation M of R can be a upper triangle matrix, it has the bigggest oder \frac{n(n+1)}{2}.
    (b)Since the element in the diagonal of M should be 1, Thus there are 2^{\frac{n(n-1)}{2}} distinct antisymmetric relations on B having the size found in (a)
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  3. #3
    MHF Contributor
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    The same problem was also considered here. If you have difficulties, I suggest starting with writing some examples of such relations with the maximum number of pairs for small m.
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