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Max, min, least upper, and greatest lower bound for sets question

  1. Oct 23, 2004 #1
    Hi,

    I was wondering if I am doing this correctly. The question asks to state the maximum value, minimum value, least upper bound, and greatest lower bound of a bunch of given sets.

    The question I am asking for is this one. {x : x E (0, 1)}

    I am a bit confused. Therefore, is the following correct?


    Maximum value: 0.9
    Minimum value 0.1
    Least Upper Bound: I'm not sure
    Greatest Lower Bound: Not sure too.

    For the least upper bound and greatest lower bound, how do I prove that IT IS the least upper bound and greatest lower bound?

    Thanks in advance. Sorry for the stupid question. I'm slowly getting a hang of University math...
     
  2. jcsd
  3. Oct 23, 2004 #2

    Hurkyl

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    0.9 isn't the maximum value; it's smaller than 0.95.
     
  4. Oct 23, 2004 #3
    Remember this, a set of real numbers does not necessarilly have a maximum or minimum but if it is bounded it always has a greatest lower bound and a least upper bound. When looking for the lub (or the glb) make sure to check if it is a bound for the set and then make sure that no smaller bound will suffice.
     
  5. Oct 23, 2004 #4
    The maximum of a set is the largest value in the set; however in this case you can show that there is no maximum. Given any value in the set, you can always find a larger value that is also in the set.


    The least upper bound is the smallest number that is larger than every number in the set; to show that 'x' is the least upper bound of a set, you thus have to show two things:

    1) show that x is larger than every element of the set (in other words, show that x is an upper bound for the set)

    2) show that if y < x, then there must be some element of the set which is larger than y (show that no number smaller than x is also an upper bound)
     
  6. Oct 23, 2004 #5

    arildno

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    Hint: (Arithmetic) averages are useful here..
     
  7. Oct 23, 2004 #6

    HallsofIvy

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    Alright, Arildno, I'll bite. WHY would "arithmetic averages" be useful here?
     
  8. Oct 23, 2004 #7

    arildno

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    Let for example a point "x" be in the set; we'll show that there exist a point y in the set so that
    x<y<1
    (that is, there is no maximum)
    just set y=(x+1)/2, this is clearly within the set (0<y<1), in addition, we have x<y
     
  9. Oct 23, 2004 #8

    quasar987

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    I'm also learning this stuff right now Kata. We just had our first exam least week. I'm guessing the mean for the group will be somewhere around 25%.

    I also have a question (more like a confusion really) regarding your question.

    We know, according to Arildo's argument that the Least Upper Bound is 1, because for anything less than that, there is always an x element of the set as near as we'd like to 1.

    But we also have that the number 1 is NOT an element of the set. Therefore we cannot say the 1 is the maximum value. But 0.9 periodic is part of the set, and 0.9 periodic = 1 (!). So 1 IS part of the set after all. The same goes for 0, is it also part of the set. So the writting E = {x : x E [0,1]} is an equivalent writting. Is this right ???
     
    Last edited: Oct 23, 2004
  10. Oct 23, 2004 #9

    Hurkyl

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    No it's not, because it is false that 0 < 0.999... < 1. (precisely because 0.999... = 1)
     
  11. Oct 23, 2004 #10

    arildno

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    0.9 periodic is no other number than 1.
    Hence, it is NOT in (0,1).
     
  12. Oct 23, 2004 #11

    quasar987

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    Oh ok thanks for clarifying that.

    What are the maximum and minimum values then? We never saw that "concept" in class or in the book we use.
     
  13. Oct 23, 2004 #12

    Hurkyl

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    Are you sure there are maximum and minimum values?
     
  14. Oct 23, 2004 #13

    quasar987

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    Usually when the books asks for them, they exist... :)

    But I guess not since you can always get closer to 0.9 periodic without quitte getting there.
     
  15. Oct 24, 2004 #14
    Thanks for the help. From the replies, this is what I have compiled, so is the following correct?

    Maximum Value: DNE because for every number x < 1, there will always be a number greater than x.

    Minimum Value: DNE because for every number x > 0, there will always be a number less than x.

    Least Upper Bound: 1 because there will always be an x that will be close to 1, but does not reach it, since the set does not include 1 itself.

    Greatest Lower Bound: 0 because there will always be an x that will be close to 0, but does not reach it, since the set does not include 0 itself.
     
  16. Oct 24, 2004 #15

    arildno

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    Max/min almost perfect; you should tweak your sentences into:
    "..there will always be a number WITHIN THE SET.."
    "Least Upper Bound: 1 because there will always be an x that will be close to 1, but does not reach it, since the set does not include 1 itself.

    Greatest Lower Bound: 0 because there will always be an x that will be close to 0, but does not reach it, since the set does not include 0 itself."
    These are unfortunate, in particular the "does not reach it" part.
    Rather, you should say something like (for LUB):
    1 is the least upper bound, because
    a) It is an upper bound for the set
    AND
    b) Any (positive) number strictly less than 1 is within the set, but since such a number is not the maximum value of the set , it certainly cannot be an upper bound for the set.
     
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