How do we find the least upper bound and greatest lower bound?

In summary, the conversation discussed solving inequalities and expressing the solutions in interval and set builder notation. It also mentioned finding the least upper bound and greatest lower bound for the solution sets, which may or may not be part of the set itself. Additionally, there was a question about understanding the concept of bounds.
  • #1
Charismaztex
45
0

Homework Statement



1. (a) Solve the following inequalities and express the solutions first in interval notation, then
express those intervals in set builder notation.

(i) [tex]x3 + x2 > 2x[/tex] (ii) [tex]\left|(2-x)\right| \leq 4[/tex] .

(b) For each of the solution sets in part (a), state the least upper bound and greatest lower bound,
if these exist, or say they do not exist.

Homework Equations



N/A

The Attempt at a Solution


I have found that:
For (i) x is between -2 and 0 or x greater than 1.
For (ii) x is between -2 and 6 (including -2 and 6)

Are the bounds just the extreme values of the domain that the function can take? I just want to make sure.

Thanks,
Charismaztex
 
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  • #2
Charismaztex said:
Are the bounds just the extreme values of the domain that the function can take? I just want to make sure.

Not exactly. The least upper bound and greatest lower bound need not be part of the set at all. The greatest lower bound is just the greatest value such that every element of the set is greater than it. For example, in your first problem, the greatest lower bound of the solution set is -2, but -2 is not in the domain.
 
  • #3
Thanks :)
 

1. What is the definition of least upper bound and greatest lower bound?

The least upper bound is the smallest number in a set that is greater than or equal to all other numbers in the set. The greatest lower bound is the largest number in a set that is less than or equal to all other numbers in the set.

2. How do we determine the least upper bound and greatest lower bound of a set?

To find the least upper bound, we need to first order the numbers in the set from least to greatest. Then, we look for the smallest number that is still greater than or equal to all the numbers in the set. This number is the least upper bound. To find the greatest lower bound, we follow the same process but look for the largest number that is still less than or equal to all the numbers in the set.

3. Why is it important to find the least upper bound and greatest lower bound of a set?

Finding these bounds can help us understand the range of numbers in a set and determine the limits or boundaries of a problem. It is also useful in mathematical proofs and in real-world applications such as financial analysis or optimization problems.

4. Are there any specific methods or algorithms to find the least upper bound and greatest lower bound?

Yes, there are different methods and algorithms that can be used to find these bounds, such as the binary search method or the bisection method. These methods involve repeatedly dividing the set into smaller subsets and comparing the numbers in each subset to find the desired bound.

5. Can the least upper bound and greatest lower bound be the same number?

Yes, it is possible for the least upper bound and greatest lower bound to be the same number in some cases. For example, if the set only contains one number, that number would be both the least upper bound and greatest lower bound. However, in most cases, these bounds will be different numbers.

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