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Logic and Proofs

  1. Sep 3, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove or give a counterexample to each statement.

    (S ∩ T) ∪ U = S ∩ (T ∪ U)


    3. The attempt at a solution

    If I proved by the contrapositive

    S (T ∩ U) ≠ (S ∩ T) ∪ U

    where would I go from there. How do I find the contrapositive with the unions and intersections?

    Thanks
     
  2. jcsd
  3. Sep 3, 2007 #2

    learningphysics

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    I don't see how that's a contrpositive... contrapositive involves an implication... but this is an equality you need to prove or disprove...
     
  4. Sep 3, 2007 #3
    (S ∩ T) ∪ U = S ∩ (T ∪ U)

    then:

    T∪U ∩ S = S ∩ (T ∪ U)

    but then where do I go from here
     
  5. Sep 3, 2007 #4

    learningphysics

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    How did you get that?
     
  6. Sep 3, 2007 #5
    The contrapositive doesn't work because you're not proving implication, but an equivalence of sets.

    Choose an arbitrary element contained in the set on one side of the equality, and then show that it's contained in the set on the other side of the equality. That proves the first set is a subset of the set on the other side. Then start on the other side of the equality and follow the same process. That proves equivalence.

    That's the gameplan for your proof.
     
    Last edited: Sep 3, 2007
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