Analyze finite sets or intervals with exact bounds. See lower bounds, upper bounds, and attainment. Download clean outputs for homework, revision, and classroom reference.
| Example Set | Set Type | Greatest Lower Bound | Least Upper Bound | Minimum | Maximum |
|---|---|---|---|---|---|
| {-4, 1, 7, 9} | Finite Set | -4 | 9 | -4 | 9 |
| (2, 8] | Interval | 2 | 8 | Does not exist | 8 |
| [3, 3] | Singleton Interval | 3 | 3 | 3 | 3 |
For a non-empty finite set S = {x₁, x₂, ..., xₙ}:
GLB(S) = inf(S) = min(S)
LUB(S) = sup(S) = max(S)
All lower bounds satisfy x ≤ inf(S).
All upper bounds satisfy x ≥ sup(S).
For an interval with lower endpoint a and upper endpoint b:
GLB = a
LUB = b
The minimum exists only if a belongs to the interval.
The maximum exists only if b belongs to the interval.
The greatest lower bound and least upper bound are core ideas in real analysis. They describe how a set behaves near its edge values. The greatest lower bound is also called the infimum. The least upper bound is also called the supremum. These values help students study sets, intervals, sequences, and inequalities with more confidence.
A lower bound sits below or at every value in a set. The greatest lower bound is the largest number with that property. An upper bound sits above or at every value in a set. The least upper bound is the smallest number with that property. These definitions are precise and useful in proofs.
For a non-empty finite set, the infimum always equals the minimum. The supremum always equals the maximum. That makes finite examples easier to test. Intervals need more care. In an open interval, an endpoint may control a bound without belonging to the set. Then the bound exists, but the minimum or maximum may not exist.
Students often mix up a bound with an attained value. This calculator separates those ideas clearly. It shows the greatest lower bound and least upper bound first. Then it checks whether each bound is actually reached. That distinction matters in homework, exam questions, and proof writing.
This tool supports both finite set input and interval input. It sorts values, removes duplicate entries, and reports clean set notation. It also lists whether a minimum or maximum exists. The export options help with revision notes, worksheets, and class examples. Use it to verify answers fast and build stronger intuition about bounds.
You can use greatest lower bound and least upper bound ideas in set notation, interval notation, sequence limits, optimization questions, and proof-based exercises. They also appear when checking boundedness and completeness of the real numbers. By comparing infimum, supremum, minimum, and maximum side by side, learners see exactly where a boundary value exists and where it is only approached.
The greatest lower bound may or may not belong to the set. The minimum must belong to the set. In finite non-empty sets, they are the same. In open intervals, they can differ.
The least upper bound is the smallest upper bound. The maximum is the largest element inside the set. A set can have a least upper bound without having a maximum.
Usually no. For example, (2, 8) has infimum 2 and supremum 8, but neither endpoint belongs to the set. So it has no minimum and no maximum.
Yes. Every non-empty finite set of real numbers is bounded. Its smallest value is the greatest lower bound, and its largest value is the least upper bound.
If both endpoints are included, the interval becomes a singleton set. Then the greatest lower bound, least upper bound, minimum, and maximum are all the same value.
Yes. The calculator accepts positive numbers, negative numbers, integers, and decimals. It also lets you control displayed precision for cleaner output.
No. Repeated values do not change the set bounds. The calculator removes duplicates in the final set display, because set elements are counted uniquely.
CSV is useful for spreadsheets, records, and worksheets. PDF is useful for printing, sharing, and saving a fixed summary of the calculated bounds.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.