1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Upper/Lower Bounds

  1. Aug 4, 2009 #1
    1. The problem statement, all variables and given/known data

    Let S=P{2,3,4,6,7,8,14,28,42,98} and let p be the relation on S defined by a p b iff a|b. Then (S,p) is a poset.

    (a) Find a subset of S which has no upper bound and no lower bound.
    (b) Find the least upper bound for {3,7}
    (c) Determine wether or not the subset {2,6,8} of S is totally ordered. Justify your answer.

    2. Relevant equations

    3. The attempt at a solution

    (a) So, I drew a lattice diagram:

    http://img514.imageshack.us/img514/8835/72872757.gif [Broken]

    I think the set S'={7,3,8} qualifies as a subset of S which has no upper bound and no lower bound. Am I right?

    (b) lub{3,7} = 7 ?

    (c) I'm not sure how to answer this question. I think that a "total ordering" on a set means a partial order ~ on a set with the additional property that for any [tex]a,b \in S[/tex] we have a~b or b~a. Here in the subset {2,6,8} of S, 6 is devisible by 2, 6|2, also 8|2 but 6 isn't devisible by 8 or vice versa, so the set isn't totally ordered. Is this correct?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 4, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi roam! :smile:
    (a) yes :smile:
    (b) nooo :redface:
    (c) yes :smile:
     
  4. Aug 4, 2009 #3

    Hi Tiny tim!

    I don't understand why my answer to part (b) is not correct, could you explain please? 3x7=21 which is not even in the bigger set, and the only (and hence smallest) upperbound is 7. :rolleyes:

    Also for part (c), is my explanation correct/sufficient (because the question says you must justify the answer)?
     
  5. Aug 4, 2009 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    How can 7 be the least upper bound if 3 isn't less than 7? You say 21 isn't in S, and that's fine... what numbers in S could be the least upper bound?
     
  6. Aug 4, 2009 #5

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi roam! :wink:
    Office_Shredder :smile: is right … 3 does not divide 7 (ie, is not "less than" in the ordering), so how can 7 be the lub?
    Yes … you've specifically said that "6 isn't divisible by 8 or vice versa", which itself shows that there is no total ordering … that's fine! :smile:
     
  7. Aug 5, 2009 #6
    The set has no upper/lower bound since there is no order on the set, 3 & 7 aren't devisible. So is it just the empty set i.e. lub{3,7}= {[tex]\emptyset[/tex]}? :redface:

    Tiny tim, there's another question that I'm confused about: "Find all maximal and minimal elements of S."
    It says "elements", plural. But there's only one max and one min:
    maxP{2,3,4,6,7,8,14,28,42,98}=98
    minP{2,3,4,6,7,8,14,28,42,98}=2
     
  8. Aug 5, 2009 #7

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi roam! :smile:

    (oh, and it's "divisible"! :wink:)
    Forget division … just look at that very good picture you drew … go up from 3 and 7, and where do you get? :smile:

    (btw, lub has to be an element

    if there's no lub, you just say so, you don't say it's the empty set. :wink:)

    I haven't come across this terminology before :redface:, but I assume maximal means any element that doesn't have anything "higher" …

    again, you can just read this off the picture. :smile:
     
  9. Aug 5, 2009 #8
    Thanks alot Tiny tim, I get it! :wink:

    lub=42
     
  10. Aug 5, 2009 #9

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    :biggrin: Woohoo! :biggrin:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Upper/Lower Bounds
  1. Upper and lower bounds (Replies: 0)

  2. Upper and Lower Bounds (Replies: 4)

Loading...