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Help with dimensions

  1. Feb 21, 2008 #1
    1. For subsets U1, U2, U3 of a finite set, prove that

    dim(U1+U2+U3) = dimU1 + dimU2 + dimU3 - dim(U1∩U2) - dim(U1∩U3) - dim(U2∩U3) + dim(U1∩U2∩U3)

    2. dim(U1+U2) = dimU1 + dimU2 - dim(U1∩U2)

    3. I found that U1+U2 theorem in my book, and I think I should use that, but I'm not sure where to start...
  2. jcsd
  3. Feb 21, 2008 #2


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    Let V = U1 + U2. Now apply the theorem to V + U3.

    Unless you are asked to prove 2 before proving 1. If this is the case please make it clear.
    Last edited: Feb 21, 2008
  4. Feb 21, 2008 #3


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    The two equations are also true, and easier to see, with the vectors spaces replaced by finite sets and the dimensions replaced by the sizes of the sets. It's possible, by picking certain bases, to make the two problems equivalent. But, as enumaelish hints, induction is probably easier.
  5. Feb 21, 2008 #4

    I am only asked to prove equation one, but in doing that, I am allowed to use equation 2.
  6. Feb 21, 2008 #5
    Thanks guys, I've got one more question.

    I did: dim(V+U3)
    and I've ended up with:
    dim(V+U3) = dimU1 + dimU2 - dim(U1∩U2) + dimU3 - dim(V∩U3)

    Is there a property I can use to show that dim(V∩U3) is equivalent to the terms I still need to include for the proof? I can't find anything helpful in my book...
  7. Feb 21, 2008 #6


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    [tex]dim(V\cap U_{3}) = dim((U_{1}+U_{2})\cap U_{3}) = dim((U_{1}\cap U_{3}) + (U_{2}\cap U_{3}))[/tex]

    Break that term up again using the second part and you're done.
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