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Set theory

  1. Jul 8, 2009 #1
    1. The problem statement, all variables and given/known data
    Is this statement true?

    For all sets A,B contained in a universe U, P(A) U P(B) is a subset of P(A U B) if and only if A is a subset of B or B is a subset of A
     
  2. jcsd
  3. Jul 8, 2009 #2

    Mark44

    Staff: Mentor

    What is P?
     
  4. Jul 8, 2009 #3
    It's sort of pf policy that you show some attempt at a problem before we try to help.

    you should try looking at various disjoint sets. For instance, take U = set of all integers, take A={0},B={1}.
     
  5. Jul 8, 2009 #4
    Mark, P(A) is probably the power set of A.
     
  6. Jul 8, 2009 #5
    P is the power set.
     
  7. Jul 8, 2009 #6
    this is just a part of a question. I did try doing it. Here is the actualy question:

    for sets A and B, P(A intersection B) = P(A) intersection P(B). However,
    the same property does not hold for unions. To fully investigate the corresponding
    property for unions, do the following exercise:
    Let A and B be sets contained in a universal set U.
    (a) Prove that P(A) U P(B) is a subset P(A U B).
    (b) Give examples of sets A and B, for which P(A) U P(B) is not equal to P(A U B).
    (c) Under what conditions on A and B will P(A) U P(B) = P(A U B)?
    State your answer in the form of a theorem: i.e.
    ”Theorem
    For all sets A and B contained in a universe U, P(A) U P(B) = P(A U B)
    if and only if ... ”
    (d) Prove your theorem from part (c).

    For a) I have the following: Assume x belongs to P(A) U P(B)
    Hence X is a subset of A or x is a subset of B
    Let Z belong to X
    Hence Z belongs to A or B
    hence Z belongs to A U B
    Hence x is a subset of A U B
    Hence x belongs to P(A UB)
     
  8. Jul 8, 2009 #7
    For part b) An example is A= {1,2,3}, B= {2,3,4}
    A u B ={ 1,2,3,4}
    Let x= {1,4} X is a subset of A U B but not a subset of A or B
     
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