# Would it be true that if a set is bounded

• DPMachine
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.
DPMachine
In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?

DPMachine said:
In general, would it be true that if a set is bounded, there must also be a supremum for the set? Too obvious, perhaps?
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.

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 rational number as supremum. As a set of real numbers, its supremum is $\sqrt{2}$.

## 1. Would it be true that if a set is bounded, it is also closed?

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

## 2. How do you determine if a set is bounded?

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.

## 3. Can a set be unbounded in one direction and bounded in another?

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.

## 4. Is a finite set always bounded?

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

## 5. Can a set be bounded and infinite?

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

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