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Empty Families

  1. Jun 21, 2009 #1
    I am beginning to study set theory and came across the following example:

    Let [tex]\mathcal{A}[/tex] be the empty family of subsets of [tex]\mathbb{R}[/tex]. Since [tex]\mathcal{A}[/tex] is empty, every member of [tex]\mathcal{A}[/tex] contains all real numbers. That is, [tex]((\forall A)(A\in\mathcal{A}\Rightarrow x\in A))[/tex] is true for all real numbers x. Thus [tex]\bigcap_{A\in\mathcal{A}} A = \mathbb{R}[/tex].

    My problem is with the first sentence. Since a family is simply a set of sets, If we talk about an empty family wouldn't this simply be the empty set [tex]\emptyset[/tex]? And since the empty set is defined not to contain anything, how could it contain any subsets of the set of real numbers?
     
  2. jcsd
  3. Jun 21, 2009 #2
    It does not contain anything. The second sentence is vacuously true.
     
  4. Jun 21, 2009 #3
    Actually, the intersection of the empty set is V, the class of all sets.
     
  5. Jun 21, 2009 #4

    HallsofIvy

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    Yes, that's true. "every member of [tex]\mathcal{A}[/tex] contains all real numbers" is the same as "if U is a member of [tex]\mathcal{A}[/tex] then U contains all real numbers". The statement "if A then B" is true whenever A is false, irrespective of whether B is true or false (that is what slider142 means by "vacuously true"). Since "U is a member of [tex]\mathcal{a}[/tex] is always false, anything we say about U is true!

    It doesn't. That is not what the statement says!
     
  6. Jun 21, 2009 #5
    I think I understand now. Since the intersection over [tex]\mathcal{A}[/tex] is defined as [tex]\left\{x: (\forall A)(A\in \mathcal{A} \Rightarrow x\in A)\right\}[/tex] and the antecedent of the conditional is always false (there is nothing in [tex]\mathcal{A}[/tex]), the conditional will always be true, because of the way the conditional operator is defined. So x can be anything in the universe. This seems a little backwards to my way of thinking, but I guess that's ok. I'll have to study that article on vacuous truth, it looks interesting. Thanks!

    edit: Ah, thanks HallsofIvy. I was busy editing this post while you responded.
     
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