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I can't think of a counterexample to disprove this set theory theorem

  1. Dec 17, 2011 #1
    I can't think of a counterexample to disprove this set theory "theorem"

    Assume F and G are families of sets.

    IF [itex]\cup[/itex]F [itex]\bigcap[/itex] [itex]\cup[/itex]G = ∅ (disjoint), THEN F [itex]\bigcap[/itex] G are disjoint as well.
     
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  3. Dec 17, 2011 #2

    micromass

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    Re: I can't think of a counterexample to disprove this set theory "theorem"

    Think of using the empty set efficiently.
     
  4. Dec 17, 2011 #3
    Re: I can't think of a counterexample to disprove this set theory "theorem"

    Just so I understand, an empty set in a family of sets would be the following?:
    Family of sets F = {{1,2,3},{4,5,6},{∅}}?

    Suppose I included an empty set in both of the families, would the intersection still be a disjoint or would it be the set {∅}?
     
    Last edited: Dec 17, 2011
  5. Dec 18, 2011 #4
    Re: I can't think of a counterexample to disprove this set theory "theorem"

    It doesn't hold if F = G = {[itex]\emptyset[/itex]}.
     
  6. Dec 18, 2011 #5
    Re: I can't think of a counterexample to disprove this set theory "theorem"

    So empty sets are not counted in a union, but they are in an interception?
     
  7. Dec 24, 2011 #6
    Re: I can't think of a counterexample to disprove this set theory "theorem"

    No, that's not it.
    [itex]\cup F \bigcap \cup G = \cup \{\emptyset\} \bigcap \cup \{\emptyset\}[/itex]
    [itex]= \emptyset \bigcap \emptyset[/itex]
    [itex]= \emptyset[/itex]
    However, [itex]F \cap G = \{\emptyset\}[/itex], which is a non-empty set, so F and G are not disjoint.
    (Note it doesn't really make sense to say "[itex]F \cap G[/itex] is disjoint" - it takes 2 sets to be disjoint, and [itex]F \cap G[/itex] is only a single set. So I assume you meant "F and G are disjoint" and not "[itex]F \cap G[/itex] is disjoint."
     
  8. Dec 28, 2011 #7
    Re: I can't think of a counterexample to disprove this set theory "theorem"

    Could you further clarify how the union of a family set consisting of just {∅} becomes ∅?

    From my previous example, what would ∪F be?
    Family of sets F = {{1,2,3},{4,5,6},{∅}}
    ∪F = {1, 2, 3, 4, 5, 6} or {∅, 1, 2, 3, 4, 5, 6}?
     
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