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Finding Supremum and Maximum

  1. Sep 23, 2012 #1
    1. The problem statement, all variables and given/known data
    For each subset of ℝ, give its supremum and maximum, if they exist. Otherwise, write none.


    2. Relevant equations
    d) (0,4)

    3. The attempt at a solution
    For part d, if the problem were [0,4], both the supremum and maximum would be 4, since the interval includes the end points, but I'm not sure about when it doesn't. It seems to me that the supremum would still be 4, as that qualifies as a least upper bound because 4 is in the neighborhood of the set, but is the maximum also 4? I don't think it would be since the interval does not include end points, but I'm confused!

    Thanks!
     
  2. jcsd
  3. Sep 23, 2012 #2

    LCKurtz

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    You aren't too confused. You are correct that the max is not 4. Does the max even exist?
     
  4. Sep 23, 2012 #3
    I think it does, but I don't know what to call it. Since my interval is a subset of the real numbers, there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set.

    Am I making this too complicated? Does the max have to be some integer?
     
  5. Sep 23, 2012 #4

    HallsofIvy

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    No, you are not making it too complicated, nor does the maximum have to be an integer. However, what you are saying, "there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set" is wrong. The maximum of a set is, by definition the largest number in that set. If the set were "[0, 4]" or "(0, 4]" then the maximum would be 4. But (0, 4) is the set of all numbers larger than 0 and less than 4. If "x" is in that set, what could you say about (x+ 4)/2?
     
  6. Sep 23, 2012 #5
    Since x is in the interval (0,4), that is x is less than 4, (x+4)/2 is in that interval.
     
  7. Sep 23, 2012 #6

    LCKurtz

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    Even better, it is between x and 4. Hall's point is that any number x you might use for the maximum has to be in the interval so it is less than 4. Yet (x+4)/2 is in the interval and greater than x. So....?
     
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