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Math Help - Question in cardinality

  1. #1
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    Question in cardinality

    k1, k2, m1, m2 are cardinal numbers.
    I need to prove that if k2>=k1 and m2>=m1, then k2m2>=k1m1
    I know that i need to prove that this function f:AXC->BxD is injective, but then what?
    Thanks for any help :-)
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Right Track

    I mean I think you are already pretty much done. I am guessing from your notation this is what you mean. You know since k1=|A|\geq |B|=k2 and m1=|C|\geq |D|=m2 then there must exist injective functions f_1:A \rightarrow B and f_2:C \rightarrow D. But then it is pretty obvious that f:A\times C \rightarrow B \times D defined by f((a,c))=(f_1(a),f_2(c)) would also be injective. Just write out definitions and it is clear.
    Last edited by Gamma; December 24th 2008 at 03:43 PM.
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