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If and only if Proof

  1. Sep 9, 2007 #1
    1. The problem statement, all variables and given/known data

    Prove or give a counterexample to the statement:

    S ∪ T = T ↔ S ⊆ T

    3. The attempt at a solution

    What I did:

    Let S={1,2,3,4} and T = {1,2}

    S ∪ T = {1,2} = T

    S ⊆ T

    {1,2,3,4} ⊈ {1,2}

    Therfore it is False . . .but the answer in the book says that it is true

    Thanks
     
  2. jcsd
  3. Sep 9, 2007 #2

    D H

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    You are confusing union and intersection. The intersection of {1,2,3,4} and {1,2} is {1,2} but their union is {1,2,3,4}.
     
  4. Sep 9, 2007 #3

    dextercioby

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    HINT: [itex]S\cup T\subset T [/itex].
     
  5. Sep 9, 2007 #4
    This isn't tru in general for any S and T, for example let T={1,2,3,4}, and S={1,5} then SUT={1,2,3,4,5} which is not a subset of T. It is true, however, if you replace the union with intersection.

    EDIT: It's also true if you change the direction of inclusion to say that T is a subset of SUT.
     
  6. Sep 10, 2007 #5
    You missed the point of the hint. It's true in this problem because you're given that S U T = T. It follows from the definition of equality.

    He gave you the first step to the proof. Now you have to ask what that says about the relationship between S and T?
     
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