Supremum and Infimum Input Panel
Example Data Table
| Example Type | Input | Infimum | Supremum | Minimum | Maximum |
|---|---|---|---|---|---|
| Finite set | {-4, 1, 3, 7, 7, 10} | -4 | 10 | -4 | 10 |
| Interval | (-2, 9] | -2 | 9 | Does not exist | 9 |
| Unbounded interval | [3, ∞) | 3 | ∞ | 3 | Does not exist |
Formula Used
Finite set: For a nonempty finite set S = {x₁, x₂, ..., xₙ},
inf(S) = min(S)
sup(S) = max(S)
diameter = sup(S) - inf(S)
Interval notation: If S = (a, b], [a, b), [a, b], (a, b), then
inf(S) = a and sup(S) = b
A minimum exists only when the left endpoint is included.
A maximum exists only when the right endpoint is included.
How to Use This Calculator
- Choose Finite numeric set for listed values, or Interval notation for bounds such as
(-∞, 5]. - Enter your values or interval. For finite sets, you can optionally apply inclusive lower and upper subset filters.
- Set the decimal precision, and enable duplicate removal if you want a unique-value analysis.
- Click Calculate to display the result panel directly under the page header and above the form.
- Use the CSV and PDF buttons to save the displayed summary for class notes, reports, or checking work.
Frequently Asked Questions
1. What is the difference between supremum and maximum?
The supremum is the least upper bound. The maximum is the greatest element actually inside the set. They match for every nonempty finite real set, but they can differ for open intervals.
2. What is the difference between infimum and minimum?
The infimum is the greatest lower bound. The minimum is the smallest element contained in the set. They are equal for nonempty finite sets, yet may differ when a lower endpoint is excluded.
3. Can this calculator handle repeated values?
Yes. Repeated values do not change the supremum or infimum. You may also remove duplicates before analysis to study the unique-value version of the same finite set.
4. What subset filters do?
Subset filters keep only values within the selected inclusive lower and upper limits. This helps you analyze a restricted portion of your dataset without changing the original entry order manually.
5. Why might a minimum or maximum not exist?
A minimum fails when the lower endpoint is excluded, such as in (2, 8]. A maximum fails when the upper endpoint is excluded, such as in [2, 8).
6. Does the calculator support infinity?
Yes. Interval mode accepts ∞, inf, and signed infinity forms. Infinite endpoints must stay open because infinity cannot be an included real endpoint.
7. Is the arithmetic mean needed for supremum or infimum?
No. Mean and median are supporting summaries for finite data only. The supremum and infimum depend on order and bounds, not on averaging.
8. When is this calculator most useful?
It is useful in calculus, real analysis, optimization, and classroom checking. It also helps verify interval behavior, boundedness, and whether extreme values are actually attained.