What is the Least Upper Bound Problem in Subset Inclusion?

In summary, the conversation discusses finding subsets of a given set that have a least upper bound in one set but not in another, yet still have a least upper bound in a third set. The conversation also explores the relationship between the least upper bound and the sets it is included in. A possible solution is given using the sets E, S1, S2, and S3 as defined in the conversation.
  • #1
boombaby
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Homework Statement



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.

Homework Equations





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
 
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  • #2
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?
 
  • #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:
 

1. What is the "Least Upper Bound Problem"?

The Least Upper Bound Problem, also known as the Least Upper Bound Property, is a concept in mathematics and computer science that refers to finding the smallest possible upper bound for a set of numbers or values. It is often used in the context of real numbers and is closely related to the concept of completeness.

2. How is the "Least Upper Bound" different from the "Greatest Lower Bound"?

The Least Upper Bound is the smallest possible upper bound for a set of numbers, while the Greatest Lower Bound is the largest possible lower bound for a set of numbers. In other words, the Least Upper Bound is the smallest number that is greater than or equal to all the numbers in a set, while the Greatest Lower Bound is the largest number that is less than or equal to all the numbers in a set.

3. Why is the "Least Upper Bound Property" important?

The Least Upper Bound Property is important because it allows us to define limits and continuity in calculus, and it is also a fundamental concept in order theory, which has applications in computer science and other fields. It also helps us to understand the behavior of real numbers and their relationships with each other.

4. How is the "Least Upper Bound" used in real-life applications?

The Least Upper Bound is used in a variety of real-life applications, such as in finance, where it is used to calculate the minimum amount of investment required to achieve a specific return. It is also used in engineering and physics to determine the maximum value that a system can achieve without breaking or failing.

5. What is an example of finding the "Least Upper Bound"?

An example of finding the Least Upper Bound is determining the smallest possible upper bound for the set of numbers {1, 2, 3, 4}. In this case, the Least Upper Bound is 4, as it is the smallest number that is greater than or equal to all the numbers in the set.

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