Calculator Input
Formula Used
For a finite set S = {x₁, x₂, ..., xₙ}, the calculator uses:
- inf(S) = min(S)
- sup(S) = max(S)
For an interval or union of intervals, it uses the smallest left endpoint as the infimum candidate and the largest right endpoint as the supremum candidate.
A minimum exists only if the infimum belongs to the set. A maximum exists only if the supremum belongs to the set.
When both bounds are finite, the width is:
Width = sup(S) − inf(S)
How to Use This Calculator
- Select Finite Set for listed numbers.
- Select Interval / Union for interval notation.
- Enter values with commas, spaces, or line breaks.
- Choose precision and display order.
- Enable steps if you want method details.
- Click Calculate to see the result above the form.
- Use the export buttons to save the report as CSV or PDF.
Example Data Table
| Input Type | Input | Infimum | Supremum | Minimum | Maximum | Notes |
|---|---|---|---|---|---|---|
| Finite Set | {3, 7, -1, 7, 5} | -1 | 7 | -1 | 7 | Finite set. Bounds are included. |
| Interval | (-2, 4] | -2 | 4 | Does not exist | 4 | Left endpoint is open. |
| Union | (-3, 1] U [5, 9) | -3 | 9 | Does not exist | Does not exist | Both extreme endpoints are open. |
| Half-Line | [2, ∞) | 2 | ∞ | 2 | Does not exist | Unbounded above. |
About Suprema, Infima, and Bounds
What this calculator measures
This supremum and infimum calculator helps you study ordered sets with speed and clarity. It works with finite sets, interval notation, and unions of intervals. The tool identifies the least upper bound and greatest lower bound. It also checks whether a true minimum or maximum exists. That difference matters in proofs, homework, calculus, and real analysis. A set can have a supremum without containing it. The same idea applies to the infimum.
Why these results matter
Supremum and infimum describe the outer limits of a set. They tell you how far a set extends, even when endpoints are open. This is useful in limit problems, optimization questions, and boundedness tests. Students often confuse supremum with maximum. They also confuse infimum with minimum. This calculator separates those ideas clearly. It reports bounds, inclusion, and width. That makes abstract notation easier to understand and verify.
How interval analysis works
For a finite set, the smallest value is the infimum and minimum. The largest value is the supremum and maximum. For intervals, the endpoint values define the bounding candidates. Open brackets mean the endpoint is excluded. Closed brackets mean the endpoint is included. A union of intervals may share one global lower bound and one global upper bound. Still, a minimum or maximum may fail to exist. The calculator checks that logic automatically.
Useful for study and checking
This page is helpful for exam revision, worksheets, and proof checking. You can test decimals, negative numbers, repeated values, and unbounded sets. You can also export results for notes or class examples. The generated summary is easy to read. It is also useful for teachers building answer keys. If you need fast least upper bound and greatest lower bound results, this tool gives a dependable starting point for mathematical reasoning.
FAQs
1. What is a supremum?
A supremum is the least upper bound of a set. It is the smallest number that is greater than or equal to every element in the set.
2. What is an infimum?
An infimum is the greatest lower bound of a set. It is the largest number that is less than or equal to every element in the set.
3. Is the supremum always inside the set?
No. The supremum may lie outside the set. For example, the set (0, 1) has supremum 1, but 1 is not a member of the set.
4. Is the infimum always the minimum?
No. The infimum equals the minimum only when the lower bound is included in the set. Open endpoints often remove the minimum.
5. Can an open interval have a maximum?
No. A fully open interval like (2, 5) has supremum 5, but it has no maximum because 5 is excluded.
6. What happens for unbounded sets?
If a set extends forever upward, it is not bounded above. If it extends forever downward, it is not bounded below. The calculator shows that clearly.
7. Do repeated values change the result?
No. Repeating the same number does not change the infimum or supremum. It only changes the raw input list. Set mode can remove duplicates automatically.
8. Can I use decimals and negative numbers?
Yes. The calculator accepts decimals, negative numbers, and interval notation. It also supports infinity symbols in interval mode for unbounded sets.