Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

About definition of 'Bounded above' and 'Least Upper Bound Property'

  1. Mar 16, 2012 #1
    The definition of 'Bounded above' states that:

    If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above.

    The 'Least Upper Bound Property' states that:
    If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

    My question is that why doesn't the definition of 'Bounded Above' include E≠Φ? Is there a problem when E=Φ? If not, then why does it matter when E=Φ for the 'Least Upper Bound Property'?
     
  2. jcsd
  3. Mar 16, 2012 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi jwqwerty! :smile:
    Because Φ has no least upper bound (unless S is finite). :wink:
     
  4. Mar 16, 2012 #3

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    If we're working in [itex]\mathbb{R}[/itex], then we sometimes use the convention

    [tex]\sup \emptyset =-\infty[/tex]

    But we should be careful because [itex]-\infty[/itex] is NOT a real number. The above is not an equality of real numbers, but merely a notation.

    But don't ever use that notation in class unless your instructor uses it.
     
  5. Mar 16, 2012 #4

    mathman

    User Avatar
    Science Advisor
    Gold Member

    There is something missing in the statement of the existence of least upper bound. Example: Let S = set of all rational numbers and let E = set of all rational numbers < √2, then E is bounded, but supE is not in S.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: About definition of 'Bounded above' and 'Least Upper Bound Property'
  1. Least upper bound (Replies: 5)

  2. Least upper bound axiom (Replies: 18)

Loading...