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Infimun and supremum of empty set

  1. May 3, 2009 #1
    Why do we define(by convention) that infimum of an empty set as [tex]\infty[/tex] and supremum as [tex]-\infty[/tex]?
     
    Last edited: May 3, 2009
  2. jcsd
  3. May 3, 2009 #2

    Hurkyl

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    It's not a convention -- it follows directly from the definition of the supremum as the least upper bound and the infimum as the greatest lower bound.
     
  4. May 3, 2009 #3

    matt grime

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    Remember that we say M is an upper bound for X if for all x in X.... so if X is the empty set then this is never true. Now, "false implies true is true", i.e. all possible real numbers are upper bounds for the the empty set.
     
  5. May 5, 2009 #4
    Thanks Hurkyl and matt grime for your replies. Yes I got it now!
     
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