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Math Help - Kuratowski Induction

  1. #1
    Super Member Showcase_22's Avatar
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    Kuratowski Induction

    Show that for any E \subset A, the following are equivalent:

    a). E \in K for every Kuratowski-Inductive K \subset \mathcal{P}(A).
    b). E is finite.
    I've given this a shot, but I think it's a bit jumbled. I decided to try induction:

    The first Kuratowski-Inductive subset of \mathcal{P}(A) is when K=\{\phi \}. This is finite since |K|=1.

    Assume each of the next n-1 sets formed by unions with elements x_1, \ldots , x_{n-1} \in A are finite.
    ie. \{ \phi \} \bigcup \{x_1\} \bigcup \ldots \bigcup \{x_{n-1} \}=\{\phi, \ x_1, \ldots, \ x_{n-1} \}.

    If we add another x_n \in A we get:

    \{\phi, x_1, \ldots , \ x_{n-1} \} \bigcup \{x_n \}=\{\phi, x_1, \ldots , \ x_{n} \}.

    This is finite since there are n+1 elements in this set.

    Is this right?
    Also, do I need to prove the converse? My notes tell me that the converse is by definition.
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  2. #2
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    Quote Originally Posted by Showcase_22 View Post
    Kuratowski-Inductive subset
    The problem depends on what your previous definitions of 'finite' and 'Kuratowski-inductive' are.

    And it would help to know what previous theorems have been established.

    What book are you working from? Suppes maybe? What exercise number?

    What is your definition of 'K is Kuratowski-inductive'?

    Do you mean [here 'u' is used for binary union]?:

    if x subset of PK &
    0 in x &
    for all y in K, we have {y} in x &
    for all b and c in x, we have buc in x
    then, x = PK

    I don't know whether that is what 'Kuratowski-inductive' means, but if it does, then K is finite iff K is Kuratowski-inductive.
    Last edited by MoeBlee; March 9th 2011 at 01:19 PM.
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