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Least upper bound problem

  1. Sep 22, 2008 #1
    1. The problem statement, all variables and given/known data

    Find subsets E[tex]\subset[/tex]S1[tex]\subset[/tex]S2[tex]\subset[/tex]S3[tex]\subset[/tex]Q such that E has a least upper bound in S1, but does not have any least upper bound in S2, yet does have a least upper bound in S3.

    2. Relevant equations



    3. The attempt at a solution

    I got totally stuck with it. If a[tex]\in[/tex]S1 is the least upper bound of E, does that mean a is also in S2 and hence a least upper bound of E in S2?
    On the other hand, if E has a least upper bound b in Q, is b unique? So in any subset S of Q, just check if b is in S to see if E has a least upper bound in S?
    Thanks a lot
     
  2. jcsd
  3. Sep 22, 2008 #2

    Dick

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    I'll get you started. Let E=[0,1) and S1=union(E,{2}). E has a LUB in S1 of 2. Do you see how the LUB can depend on the set E is included in? Can you finish?
     
  4. Sep 22, 2008 #3
    O I got it now....So S2=union(S1,{x[tex]\in[/tex]Q|x^2>2, 0<=x<=2}) would work. And S3=union(S2,{1.1})
    Thanks very much:rofl:
     
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