What is the Least Upper Bound Problem in Subset Inclusion?

  • Thread starter Thread starter boombaby
  • Start date Start date
  • Tags Tags
    Bound Upper bound
Click For Summary
SUMMARY

The Least Upper Bound (LUB) problem in subset inclusion involves identifying subsets E, S1, S2, and S3 within the rational numbers Q. In this discussion, E is defined as [0,1) and S1 as the union of E and {2}, where E has a LUB of 2 in S1. However, in S2, which is the union of S1 and the set of rational numbers x where x^2 > 2 and 0 ≤ x ≤ 2, E does not have a LUB. Conversely, in S3, which includes an additional element 1.1, E does have a LUB. This illustrates how the LUB can vary based on the subset in which E is included.

PREREQUISITES
  • Understanding of least upper bounds in real analysis
  • Familiarity with set theory and subset notation
  • Knowledge of rational numbers and their properties
  • Basic concepts of unions of sets
NEXT STEPS
  • Study the properties of least upper bounds in real analysis
  • Explore set theory, focusing on unions and subsets
  • Investigate examples of least upper bounds in various subsets of rational numbers
  • Learn about the completeness property of the real numbers
USEFUL FOR

Students in mathematics, particularly those studying real analysis, set theory, or anyone tackling problems related to least upper bounds and subset inclusion.

boombaby
Messages
129
Reaction score
0

Homework Statement



Find subsets E\subsetS1\subsetS2\subsetS3\subsetQ 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\inS1 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
 
Physics news on Phys.org
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?
 
O I got it now...So S2=union(S1,{x\inQ|x^2>2, 0<=x<=2}) would work. And S3=union(S2,{1.1})
Thanks very much:smile:
 

Similar threads

Least Upper Bound Property ⇒ Archimedean Principle
  • · Replies 1 ·
Replies
1
Views
2K
Set of least upper bounds multiplied by a constant
  • · Replies 1 ·
Replies
1
Views
1K
Can anyone check my proof involving least-upper-bounds?
  • · Replies 33 ·
2
Replies
33
Views
3K
Ordered relations, lower upper bounds of a set
  • · Replies 4 ·
Replies
4
Views
2K
Analysis - Upper and Lower Bounds
  • · Replies 4 ·
Replies
4
Views
2K
Prove that an upper bound a is the least upper bound
  • · Replies 11 ·
Replies
11
Views
4K
Rational numbers, supremum (Is my proof correct?)
  • · Replies 6 ·
Replies
6
Views
2K
Real Analysis Least Upper Bound Question
  • · Replies 6 ·
Replies
6
Views
3K
Finding an upper bound for a contour integral (Complex)
  • · Replies 2 ·
Replies
2
Views
4K
Prove that x is in S or x is an accumulation point of S
  • · Replies 25 ·
Replies
25
Views
3K