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I Dedekind Cuts

  1. Jan 27, 2017 #1
    I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

    ##x \in Q## and ##y<x \Longrightarrow y \in Q##

    Does that mean that this is not a cut because it is bounded below?

    {## x \in Q : x>1 \wedge x<2 ##}

    Clear this up for me please.
     
  2. jcsd
  3. Jan 28, 2017 #2

    stevendaryl

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    No, it's not a cut. Another way to think of it is that a cut splits the set of all rationals into two nonempty pieces: A "left" set, [itex]L[/itex], and a "right" set, [itex]R[/itex], where
    • Every rational [itex]x[/itex] is either in [itex]L[/itex] or [itex]R[/itex].
    • If [itex]x[/itex] is in [itex]L[/itex], and [itex]y[/itex] is in [itex]R[/itex], then [itex]x < y[/itex]
    • The real associated with the pair [itex]L,R[/itex] is the unique number [itex]r[/itex] that is greater than or equal to every element of [itex]L[/itex] and less than or equal to every element of [itex]R[/itex]
    A Dedekind cut is just such an [itex]L[/itex].
     
  4. Feb 13, 2017 #3
    stevendaryl wrote:

    The real associated with the pair ##L,R## is the unique number ##r## that is greater than or equal to every element of ##L## and less than or equal to every element of ##R##

    Since Dedekind cuts are used to construct the reals I think it would be better to say that the real number ##r## is the cut, or preferably that ##L## as defined by OP is an extended real number.
     
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