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Math Help - Proof involving power set and union

  1. #1
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    Proof involving power set and union

    Hello guys,
    I need help on this question.

    Let A and B be sets. Let A = {Sk | k ∈ I } be a family of sets with I not equal to an empty set. Prove the statement.
    If P(A U B) ⊆ P(A) U P(B) then A ⊆ B or B ⊆A

    What I have in mind right now::
    Assume
    P(A U B) ⊆ P(A) U P(B)
    Let X
    P(A U B) which X ∈ A or X ∈ B
    demonstrate 2 the cases.
    Case 1:
    X ∈ A
    From the assumption Show A ⊆ B
    Case 2: X
    ∈ B
    From the assumptionShow B ⊆A
    therefore A ⊆ B or B ⊆A

    I might be wrong, but that is all i have right now and
    i don't know how to express the proposition
    P(A) U P(B) or how to use it.
    if anyone can explain it, that would be awesome.


    Any help would be appreciated.
    Thank you.
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  2. #2
    MHF Contributor

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    Re: Proof involving power set and union

    I would prove the contrapositive, show that if there is x in A-B, AND y in B-A, that then there is an element of P(AUB) that does not lie in P(A)UP(B).

    Hint: consider the set {x,y}.
    Thanks from kcyw0515
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  3. #3
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    Re: Proof involving power set and union

    Thank you. I got it.
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