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Intersections and Unions of powersets confusing

  1. Oct 5, 2009 #1
    1. The problem statement, all variables and given/known data
    let A and B be finite sets

    Let P(A) denote the Power Set of A (set of all subsets of A)


    2. Relevant equations

    Prove or disprove:
    a) P(A inter B) = P(A) inter P(B)
    b) P(A U B) = P(B) U P(B)

    3. The attempt at a solution

    So I tried drawing Venn diagrams and came to the conclusion that they must be true but when I asked my professor for help all he said was that Venn diagrams don't count as formal proofs and that I was wrong in any case. Now I am just totally confused. Any help will be appreciated
     
    Last edited: Oct 5, 2009
  2. jcsd
  3. Oct 5, 2009 #2

    Office_Shredder

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    Just try doing it with words. You really want to prove two things for each question

    a) Prove that P(a intersect B) is a subset of P(A) intersect P(B). Also prove that P(A) intersect P(B) is a subset of P(A intersect B).

    So how do we do this? For the first direction: Suppose X is an element of P(A intersect B). Your goal is to conclude that X is an element of both P(A) and P(B)
     
  4. Oct 5, 2009 #3
    ok so if x is an element of P(A inter B)
    then a is a subset of (A inter B)
    so a is a subset of A AND a is a subset of B
    so a is an element of P(A) AND a is an element of P(B)
    therefore a is an element of P(A) inter P(B).

    Is that reasoning sound?
     
  5. Oct 5, 2009 #4

    Office_Shredder

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    Yeah looks good to me.
     
  6. Oct 6, 2009 #5
    Ok thanks alot,

    Now for part b, after thinking about it I can intuitively see why the equality is false but I don't know how to prove it. Could you point me in the right direction please?
     
  7. Oct 6, 2009 #6
    OK I actually just thought that since it's easier to prove that something is false I used a counter example.
    Consider A = {1} and B = {2}
    In this case P(A) U P (B) does not contain {1, 2} but P(A U B) does. That seems logical to me..
     
  8. Oct 6, 2009 #7

    Office_Shredder

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    For part a you need to do the other direction that P(A) intersect P(B) is a subset of P(A intersect B)

    Part b sounds good also
     
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