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Math Help - How to prove distributivity of cardinals?

  1. #1
    Junior Member
    Joined
    Mar 2010
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    How to prove distributivity of cardinals?

    I have this kind of problem:
    (1): Let k,m,l be cardinals. Prove that k(m+l)=km+lm.

    I know, that I have to prove that there exist a bijection between Kx(LuM) and
    (KxL)u(KxM) where card(K) = k, card(L) = l and card(M) = m, but how I do it?
    I'm little bit lost here, can anyone help?
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  2. #2
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
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    Pick representative sets K, L and M, respectively (all mutually disjoint). You want to show that K(LM) ∼ (KL)∪(KM).
    These are actually equal since if (a,b) is in the LHS then either bL or b ∈ M. In the first case, (a,b) ∈ KL and in the second (a,b) ∈ KM.
    The converse also applies.
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