Calculator Inputs
Choose a finite set or an interval. The calculator finds the least upper bound, checks extrema, and evaluates an optional candidate upper bound.
Example Data Table
| Case | Input | Least Upper Bound | Maximum Exists? | Reason |
|---|---|---|---|---|
| Finite set | {2, 5, 5.5, 7, 9} | 9 | Yes | A finite nonempty set has supremum equal to its maximum. |
| Open interval | (1, 8) | 8 | No | The right endpoint bounds the set but is not included. |
| Half-open interval | [1, 8) | 8 | No | The least upper bound is 8, yet 8 is excluded. |
| Closed interval | [1, 8] | 8 | Yes | The right endpoint is included, so it is also maximum. |
| Unbounded interval | (-∞, ∞) | Does not exist | No | No real number can bound the whole set above. |
Formula Used
Definition: A number u = sup(S) if two conditions hold.
- Upper bound condition: every element of S satisfies x ≤ u.
- Least condition: for every ε > 0, some x ∈ S satisfies u - ε < x ≤ u.
Finite nonempty sets: if S = {x₁, x₂, ..., xₙ}, then sup(S) = max(S).
Intervals: for bounded-above intervals such as (a,b), [a,b), (a,b], or [a,b], the least upper bound is the right endpoint b.
Maximum test: the maximum exists exactly when the least upper bound belongs to the set itself.
How to Use This Calculator
- Select Finite set or Interval.
- For a finite set, enter real numbers separated by commas, spaces, line breaks, or semicolons.
- For an interval, enter endpoints, choose open or closed brackets, and mark unbounded sides when needed.
- Optionally provide a candidate upper bound to verify whether it bounds the set and whether it is least.
- Press Calculate Least Upper Bound to see the result table and the Plotly graph above the form.
- Use the export buttons to save a CSV or PDF summary of the computed report.
FAQs
1. What is a least upper bound?
A least upper bound is the smallest real number that is still at least as large as every element in a set. It is also called the supremum.
2. Is the least upper bound always inside the set?
No. In an open interval like (1, 8), the least upper bound is 8, but 8 is not a member of the interval.
3. How is supremum different from maximum?
The maximum must belong to the set. The supremum only needs to bound the set above and be the smallest such bound.
4. Why does a finite nonempty set always have a least upper bound?
A finite nonempty real set has a largest element after sorting or comparing entries. That largest element is both the maximum and the least upper bound.
5. What happens for an unbounded interval?
If the set is unbounded above, no real number can bound all elements from above. In that case, no real least upper bound exists.
6. Can the empty set have a least upper bound?
Within ordinary real-number analysis, the empty set does not receive a least upper bound from this calculator. The report flags it explicitly as empty.
7. What does the candidate upper bound test do?
It checks whether your chosen value is at least every set element. If it is also the smallest such value, the calculator labels it the least upper bound.
8. Why is the graph useful?
The graph places the set on a real line and marks the supremum visually. This makes endpoint inclusion, maximum status, and boundedness easier to interpret.