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Math Help - Bijective function proof

  1. #1
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    Bijective function proof

    Hey guys, just a bit lost with this proof

    Let x and y be distinct numbers in the set {1, 2, . . . , n}, (n positive integer) and let  \alpha \in Sn be the transposition (x y). Define
    W = {  \beta \in Sn | \beta(x) > \beta(y) },
    Z = { \beta \in Sn | \beta(x) < \beta(y) }.

    Prove that W and Z are complementary subsets of Sn and that f( \beta) = \beta\alpha defines a bijective function from W to Z.
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  2. #2
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    Hi Mr.

    let a = beta(x), b=beta(y).

    Then a neq b and 1\le a ,b \le n.

    If a < b then beta is in Z. If a < b then beta is in W. Thus
    W and Z partition S_n.

    Note that beta alpha(x) = beta(y) = b
    and beta alpha(y) = beta(x) = a.
    Thus,
    beta in W
    iff
    beta(x) > beta(y)
    iff
    a > b
    iff
    beta alpha(y) = a > b = beta alpha(x)
    \iff
    beta alpha in Z

    Thus, the map f that takes beta to beta alpha sends
    everything in W to Z.
    Since f applied twice is the identity, it follows that f also sends
    everything in Z back to W.
    I.e., f is a bijection between W and Z.

    Best,

    ZD
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  3. #3
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    thank you very much
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