What is the Least Upper Bound Problem in Subset Inclusion?

[boombaby]

Homework Statement

Find subsets E$$\subset$$S1$$\subset$$S2$$\subset$$S3$$\subset$$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.

Homework Equations

The Attempt at a Solution

I got totally stuck with it. If a$$\in$$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

 

[Dick]

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?

 

[boombaby]

O I got it now...So S2=union(S1,{x$$\in$$Q|x^2>2, 0<=x<=2}) would work. And S3=union(S2,{1.1})
Thanks very much