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About least upper bound and ordered set

  1. Mar 13, 2012 #1
    I have two questions:

    1. Least Upper Bound
    ex)
    Let A = {x : x∈Q(rational number) , x^2<2, x<0} ⊂ S (ordered set),
    Then why doesn't the least upper bound (α,which is √2) exist in S?
    As long as I know, S is a set which has an order. And it doesn't state whether it belongs to Q(rational number) or R(Real Number)

    2. Ordered Set (S)

    Whenever I see definition related to least upper bound, greatest lower bound and so forth, I always see 'A⊂S'

    For example, let's look at the defintion of 'bounded above' :
    Supposed S is an ordered set and A⊂S. If there exists a β∈S such that x≤β for every x∈A, we say that A is bounded above.

    So my questions is that why is it important to state A⊂S
     
  2. jcsd
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