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Math Help - Axiom Question

  1. #1
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    Axiom Question

    Any ideas?

    Show that the set of all x such that x is in A and x is not in B exists.
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  2. #2
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    Two things come immediately to mind:

    1) Is B not A?
    2) Define "exists". "Contains at least one element"?
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  3. #3
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    Quote Originally Posted by qwe123 View Post
    Any ideas?

    Show that the set of all x such that x is in A and x is not in B exists.
    Define the property statement P(x) to mean "x is not in B".
    By the Axiom Schema of Comprehension there is a set S such that x is S if and only if x in A and P(x) is true, that is, x is in A and x is not in B.
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  4. #4
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    Hi,
    what are A,B? sets?
    if they are sets, then the axiom schema of restricted comprehension, namely the axiom \forall b\forall a \exists z \forall x (x\in z \leftrightarrow (x \in a\,\&\, \phi(x,b))) where \phi(x,b) is the formula  x \not \in b, ensures existence of the set you described (just evaluate a to be A and b to be B)
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