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Upper bound and lower bound

  1. Jun 15, 2013 #1
    At:

    http://en.wikipedia.org/wiki/Upper_and_lower_bounds

    in example it says that

    "2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not"


    Why "2"? as 2 is not in that set.

    Also,
    at:

    http://en.wikipedia.org/wiki/Supremum

    in example it says that
    "The "Supremum" or "Least Upper Bound" of the set of numbers 1, 2, 3 is 3. Although 4 is also an upper bound, it is not the "least upper bound" and hence not the "Supremum"."


    Why? as 4 is not in the set of 1,2,3 but if for a moment I think that as 4>3 so it is the upper bound of the set which contains 1,2,3 then am I correct to say that 3 is the least upper bound ?

    Thanks in advance.
     
  2. jcsd
  3. Jun 15, 2013 #2

    micromass

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    A lower bound of a set A is any number x such that x<a for any ##a\in A##.
    So we don't need the number to be in the set (if the element is in the set, then it's called a minimum). A lower bound is just any number smaller than each element in the set.
     
  4. Jun 15, 2013 #3
    So 5 is the greatest lower bound.
     
  5. Jun 15, 2013 #4

    micromass

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    Yes. Anything lower than 5 is also a lower bound.
     
  6. Jun 15, 2013 #5
    ImageUploadedByPhysics Forums1371326706.331826.jpg

    (Copy/pasted from wiki)

    Question: as Q doesn't have the least upper bound as 2^1/2 is irrational but the example says that Q has an upper bound, is that upper bound any number greater than 2^1/2 or is it a specific number?
     
  7. Jun 15, 2013 #6
    Thank you so much.
     
  8. Jun 15, 2013 #7

    micromass

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    Any rational number greater than ##\sqrt{2}## is an upper bound.
     
  9. Jun 15, 2013 #8
    Once again, thank you very much sir.
     
  10. Jun 15, 2013 #9

    HallsofIvy

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    In fact, because 5 is in the set, 5 is the minimum of the set.
    (If a set has a minimum (smallest member) then that minimum is the greatest lower bound.) But as long as a set has lower bounds, it has a greatest lower bound whether is has a minimum or not.
     
  11. Jun 15, 2013 #10

    micromass

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    Of course, that is only true in ##\mathbb{R}##.
     
  12. Jun 15, 2013 #11

    HallsofIvy

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    No, it does not say that Q has an upper bound! It says that Q intersect the interval from [tex]-\sqrt{2}[/tex] to [tex]\sqrt{2}[/tex] has upper bounds. 1.5, for example is an upper bound of that set.

    (But Q, the set of all rational numbers, does NOT have either upper or lower bounds.)
     
  13. Jun 16, 2013 #12
    What does upper bound, least upper bound (supremum), lower bound, and greatest lower bound (infimum) tells us intutively? It just tells us lower & greater numbers, right?

    Is maximum (max) is just an other word for least upper bound (supremum), and similarly minimum (min) is just an other word for greatest lower bound (infimum)?
     
  14. Jun 16, 2013 #13

    micromass

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    Yes, the upper and lower bound just tells us lower and greater numbers. The greatest lower bound also tells us a lower number, but the best possible one.

    Not exactly. A minimum of a set A is an infimum that also belongs to the set.
    For example, 1 is an infimum of (1,2], but not a minimum since 1 does not belong to the set. On the other hand, 1 is a minimum of [1,2] and thus also an infimum.
     
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