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Math Help - Axiomatic Set theory

  1. #1
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    Axiomatic Set theory

    1. Given the following sets:
    x = {a, b}
    y= {a,d}}

    Write the Axiom of Extensionality in expanded form using conjunctions and
    disjunctions.
    Does the axiom hold true for this example?
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  2. #2
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    You have too many braces for y. Did you mean y= {a, d}?

    An axiom is always true. Do you mean to ask if the hypotheses of the axiom are true- that is, whether the axiom is applicable here.
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  3. #3
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    Quote Originally Posted by aaronrj View Post
    1. Given the following sets:
    x = {a, b}
    y= {a,d}}

    Write the Axiom of Extensionality in expanded form using conjunctions and
    disjunctions.
    Does the axiom hold true for this example?
    Do you mean {a d} ?

    Do you mean write the axiom of extensionality with the sentential connectives being conjunctions, disjunctions, and negation only?

    The ordinary axiom of extensionality is:

    Axy(Az(z in x <-> z in y) -> x = y)

    Axy(Az((z in x -> z in y) & (z in y -> z in x)) -> x = y)

    Axy(Az((z not in x or z in y) & (z not in y or z in x)) -> x = y)

    Axy(Az(not((z not in x or z in y) & (z not in y or z in x))) or x = y)

    I guess there's some point to that exercise?

    Then as to "the axiom holding true", do you mean, whether, under the axiom, we have {a b} = {a d}. Assuming that a, b, and d are all distinct from one another, then of course {a b} is not equal to {a d}, since b in {a b} and not in {a d} (as well as, d in {a d} and not in {a b}). But we don't need the axiom of extensionality for that, since it follows from idenity theory alone (given, of course, the ordinary definition of {. .} and as enabled by the pairing axiom).
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