- #1

DPMachine

- 26

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

In summary, the question is whether a set, specifically a set of real numbers, that is bounded must also have a supremum. This is generally true for the real numbers, but may not be true for other types of sets, such as rational numbers. The supremum is the smallest upper bound of a set and is a defining property of the real numbers.

- #1

DPMachine

- 26

- 0

Physics news on Phys.org

- New quantum error correction method uses 'many-hypercube codes' while exhibiting beautiful geometry
- Researchers advance new class of quantum critical metal that could advance electronic devices
- Researchers make sound waves travel in one direction only, with implications for electromagnetic wave technology

- #2

HallsofIvy

Science Advisor

Homework Helper

- 42,988

- 975

Define your terms! A "set" does not even have to consist of numbers. In order to talk about a set being "bounded" there must be some kind of metric defined on it but even then there may not be an order- the set of all complex numbers with norm less than 1 is bounded but is not an ordered set and so "supremum" makes no sense.DPMachine said:

I assume you are talking about a set of real numbers. Yes, any set of real numbers, having an upper bound, has a real number as supremum. That's one of the "defining" properties of the real numbers.

It is NOT true, for example, if you are talking about rational numbers. For example, the set of all rational numbers, x, such that \(\displaystyle x^2< 2\) has such numbers as 1.5, 2, etc. as upper bounds but does not have a

No, a set can be bounded without being closed. For example, the set (0,1) is bounded but not closed.

A set is bounded if there exists a number M such that the absolute value of every element in the set is less than or equal to M. In other words, the set is contained within a finite interval.

Yes, a set can be unbounded in one direction and bounded in another. For example, the set (0, ∞) is unbounded in the positive direction but bounded in the negative direction.

Yes, a finite set is always bounded since there exists a maximum and minimum value within the set.

Yes, a set can be bounded and infinite. For instance, the set of all real numbers between 0 and 1 is bounded but infinite.

- Replies
- 6

- Views
- 2K

- Replies
- 3

- Views
- 1K

- Replies
- 1

- Views
- 1K

- Replies
- 1

- Views
- 1K

- Replies
- 2

- Views
- 1K

- Replies
- 3

- Views
- 1K

- Replies
- 5

- Views
- 2K

- Replies
- 10

- Views
- 3K

- Replies
- 9

- Views
- 1K

- Replies
- 5

- Views
- 1K

Share: