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A set where inf(B) ≥ sup(B)

  1. Jan 17, 2017 #1

    RJLiberator

    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    Find an example set, B where inf B ≥ sup B.


    2. Relevant equations
    For reference, the definitions https://en.wikipedia.org/wiki/Infimum_and_supremum :
    "In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.

    The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB)."

    3. The attempt at a solution

    I have been reading that if we take the empty set we find that inf(empty set) = ∞ and suprema(empty set) = -∞.

    While this is not intuitive, it makes sense after some observations.

    If I define my set as the empty set, I have suceeded in finding a set where inf B ≥ sup B.

    Correct?

    Outside of this, I would think if I had the set B = {4} that the inf(B) = 4 and sup(B) = 4. Thus, inf B ≥ sup B. Correct?
     
  2. jcsd
  3. Jan 17, 2017 #2

    fresh_42

    Staff: Mentor

    Yes, this is correct. The reason is that every statement about the elements of the empty set is true. In this case all numbers are upper and lower bounds for them, or as I like to say: the elements of the empty set have purple eyes. Your statement about ##\{4\}## is trivially true as well.
     
  4. Jan 17, 2017 #3

    RJLiberator

    User Avatar
    Gold Member

    Excellent, I thank you.
     
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