# About definition of 'Bounded above' and 'Least Upper Bound Property'

• jwqwerty
In summary, the definition of 'Bounded above' states that if a set E is contained in an ordered set S, there exists a β in S such that all elements of E are less than or equal to β. The 'Least Upper Bound Property' states that if E is a non-empty, bounded set in an ordered set S, then the least upper bound of E exists in S. However, this definition does not include the condition that E cannot be empty, leading to potential issues when E is an empty set.
jwqwerty
The definition of 'Bounded above' states that:

If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above.

The 'Least Upper Bound Property' states that:
If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

My question is that why doesn't the definition of 'Bounded Above' include E≠Φ? Is there a problem when E=Φ? If not, then why does it matter when E=Φ for the 'Least Upper Bound Property'?

hi jwqwerty!
jwqwerty said:
The 'Least Upper Bound Property' states that:
If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

Because Φ has no least upper bound (unless S is finite).

If we're working in $\mathbb{R}$, then we sometimes use the convention

$$\sup \emptyset =-\infty$$

But we should be careful because $-\infty$ is NOT a real number. The above is not an equality of real numbers, but merely a notation.

But don't ever use that notation in class unless your instructor uses it.

jwqwerty said:
The definition of 'Bounded above' states that:

If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above.

The 'Least Upper Bound Property' states that:
If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper Bound) exists in S.

My question is that why doesn't the definition of 'Bounded Above' include E≠Φ? Is there a problem when E=Φ? If not, then why does it matter when E=Φ for the 'Least Upper Bound Property'?
There is something missing in the statement of the existence of least upper bound. Example: Let S = set of all rational numbers and let E = set of all rational numbers < √2, then E is bounded, but supE is not in S.

## What does it mean for a set to be "bounded above"?

When a set is bounded above, it means that there exists a number that is larger than or equal to all elements in the set. In other words, there is a maximum value in the set.

## What is the definition of "least upper bound property"?

The least upper bound property is a mathematical concept that states that every non-empty set of real numbers that is bounded above must have a least upper bound, which is the smallest number that is greater than or equal to all elements in the set.

## How is the least upper bound property used in analysis?

The least upper bound property is an important concept in analysis because it allows us to prove the existence of limits, continuity, and other fundamental properties of real numbers. It also serves as the basis for the concept of completeness in the real number system.

## Can you give an example of a set that is bounded above but does not have a least upper bound?

Yes, the set of all rational numbers less than 1 is bounded above by 1, but it does not have a least upper bound because there is always a rational number that is closer to 1 but not equal to it. In other words, there is no smallest number in this set that is greater than or equal to all elements.

## How does the concept of "bounded above" relate to the concept of "bounded below"?

The concept of "bounded above" is the opposite of "bounded below." A set is bounded below when there exists a number that is smaller than or equal to all elements in the set. A set can be both bounded above and bounded below, in which case it is called a bounded set.

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