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Intersection with empty index set

  1. Mar 15, 2008 #1
    If I is empty, and the collection of sets {A_i} is indexed by I, then the intersection of all the A_i is equal to the universal set.

    Can someone explain why? Or better yet, give a proof?
     
    Last edited: Mar 15, 2008
  2. jcsd
  3. Mar 15, 2008 #2
    Here's my attempt at a proof:

    Suppose I is empty. Let S be the universal set. Let x be in S. Then it is vacuously true that for all i in I (it is false that i is in I!), x belongs to A_i. Thus x belongs to the intersection of all the A_i. Hence the intersection of all the A_i equals S.

    Correct?


    And here's my proof attempt that U (A_i) = empty, if I is empty:

    Suppose there exists x in U (A_i). Then there exists i in I such that x belongs A_i. But I is empty, so no such i exists. This contradiction means that no such x exists. Thus U (A_i) = empty. Correct? I can't find a proof anywhere.
     
    Last edited: Mar 15, 2008
  4. Mar 15, 2008 #3
    a for all statement is not an implication statement. i think it doesn't work that way. what book is this from?
     
    Last edited: Mar 15, 2008
  5. Mar 15, 2008 #4
    Ok, let's start with U (A_i) = empty, where I is empty (I gave a proof above).

    By DeMorgan's Laws,
    intersection (A_i) = S - U (S-A_i) = S - empty = S.

    Is this a better proof? I still think my first proof may be correct.
     
    Last edited: Mar 15, 2008
  6. Mar 15, 2008 #5

    Hurkyl

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    This looks right to me.
     
  7. Mar 20, 2008 #6
    EDIT: I guess I assumed you were talking about ZFC... What theory are you talking about in particular?

    OLD:

    Well, if you define the intersection to be "the set of all things in all A_i", then intersection may not be well-defined (i.e., you're defining something that really can't exist). In particular, the comprehension axiom would require you to fix some A_j in the family {A_i} in order to be able to show that the set you call the intersection actually exists.

    If you took as an axiom that "the set of all things in all A_i" exists, you'd immediately get a Russell paradox as the OP mentioned.

    The book that I studied these topics from is Enderton's "Elements of Set Theory".
     
    Last edited: Mar 20, 2008
  8. Mar 20, 2008 #7

    Hurkyl

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    The opening poster sounded like he was talking about working in a fixed universe of discourse.


    Incidentally, even in ZF, you can take an empty intersection; it would simply be the (proper) class of all sets.
     
  9. Mar 20, 2008 #8
    That's interesting. How do you define the intersection then, in ZF? The resources that I've learned from define the intersection of a set to be the set given by the comprehension axiom. Are you specifically defining the empty intersection to be the class of all sets?
     
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