Calculator Input
Example Data Table
| Case | Input | Greatest Lower Bound | Minimum Exists | Note |
|---|---|---|---|---|
| Finite Set | {8, 3, 5, -2, 11} | -2 | Yes | For finite sets, the infimum equals the minimum. |
| Closed Interval | [2, 9] | 2 | Yes | The left endpoint belongs to the interval. |
| Open Interval | (2, 9) | 2 | No | The infimum exists, but 2 is not included. |
| Half-Open Interval | (-4, 7] | -4 | No | The left endpoint is the greatest lower bound. |
Formula Used
A number c is a lower bound of a set S if c ≤ x for every x ∈ S.
glb(S) = inf(S), the largest number that still remains a lower bound of S.
If S = {x₁, x₂, ..., xₙ} is finite and nonempty, then glb(S) = min(S).
For intervals like (a, b), [a, b], (a, b], or [a, b), the greatest lower bound is a. A minimum exists only when the left endpoint is closed.
How to Use This Calculator
Choose the input mode first. Use the finite set mode for listed numbers, or use the interval builder for endpoint-based inputs.
Enter a candidate lower bound if you want the calculator to test whether that value is a lower bound and whether it matches the greatest lower bound.
Select the desired decimal precision. Then click the calculate button to display the result summary, analysis table, export options, and graph.
Review the graph to compare the greatest lower bound with your data. Use the CSV or PDF buttons to save the result table.
FAQs
1) What is a greatest lower bound?
It is the largest value that is still less than or equal to every element of the set. In real analysis, it is also called the infimum.
2) Is the greatest lower bound always inside the set?
No. For open intervals such as (2, 9), the greatest lower bound is 2, but 2 is not included in the interval.
3) How is this different from the minimum?
A minimum must belong to the set. A greatest lower bound does not have to belong to the set. When the minimum exists, it equals the greatest lower bound.
4) What happens for a finite set?
For any nonempty finite set of real numbers, the greatest lower bound is simply the smallest element after sorting.
5) Can I test a candidate lower bound?
Yes. Enter any numeric candidate and the calculator will report whether it is a valid lower bound and whether it matches the greatest lower bound.
6) Why can an open interval still have a greatest lower bound?
Because the infimum depends on the lower-bound property, not on membership alone. The left endpoint can still be the greatest value below every interval element.
7) Does this calculator support empty inputs?
No. The tool requires a nonempty finite set or a nonempty interval. Empty inputs are rejected because this page focuses on practical real-valued examples.
8) Why is the graph useful here?
The graph helps you see how far the greatest lower bound lies below the ordered values or interval sample points. It makes infimum behavior easier to interpret visually.