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Set theory: proofs regarding power sets

  1. Jul 24, 2014 #1
    Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X).

    I really don't know how to get started on this proof but I tried to start with something like this:
    ∀ m,n ∈ A,B ⇒
    m,n ∈ X ⇒

    Is this the right way to start on this proof? A hint on how to get further with this would really be appreciated.

    Stefan
     
  2. jcsd
  3. Jul 24, 2014 #2

    verty

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    Try to use the fact that EVERY subset of X is in P(X). Can you see how that helps?
     
  4. Jul 24, 2014 #3
    @verty, thanks for your reply;

    So if A,B ∈ P(X) ⇒ A,B ⊂ X, and since every subset of X is in P(X), A∩B,A∪B are also in P(X)?

    Stefan
     
  5. Jul 24, 2014 #4

    Yes, though it wouldn't be a bad idea to show that this union and intersection are subsets of X, despite the triviality.
     
  6. Jul 24, 2014 #5
    Thanks for you reply!

    That's where it get stuck I'm afraid, I cannot link the Union and Intersection to X with the given information.

    Stefan
     
  7. Jul 24, 2014 #6
    Approach it like an introductory set theory proof.

    Pick an element x in A intersect B. Show that this is in X.

    Pick an element y in A union B. Show that this is in X.

    For the first one, suppose that x is in A intersect B. Then x is in A and x is in B. Now what?

    Remember that you know that A and B are subsets of X - they are in its power set!
     
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