Infimum Calculator

Enter values, intervals, or candidate lower bounds. Compare lower limits with clear checks and notes. Export results for reports, teaching, and quick set review.

Calculator Inputs

Separate values with commas, spaces, semicolons, or new lines.

Interval Builder

Leave a lower endpoint empty for -∞. Leave an upper endpoint empty for +∞.

Interval 1

Interval 2

Interval 3

Interval 4

Interval 5

Example Data Table

Input Type Example Set Infimum Attained? Reason
Finite values {4, 2, 9, 2.5} 2 Yes The smallest listed value belongs to the set.
Open interval (0, 6] 0 No The set approaches 0 but excludes it.
Closed interval [-3, 4) -3 Yes The lower endpoint is included.
Unbounded interval (-∞, 8) -∞ No There is no finite greatest lower bound.

Formula Used

For a finite nonempty set, the infimum equals the smallest listed value.

Finite set: inf(S) = min(S)

For interval unions, compare every nonempty interval lower endpoint.

Interval union: inf(∪Ij) = smallest lower endpoint among all nonempty intervals.

A candidate L is a lower bound when L ≤ x for every x in S.

The infimum is the greatest value among all valid lower bounds.

How To Use This Calculator

  1. Choose finite data mode or interval union mode.
  2. Enter numeric values or interval endpoints.
  3. Use -inf and inf for unbounded intervals.
  4. Select open or closed endpoint types for intervals.
  5. Add a candidate lower bound when you want verification.
  6. Choose the decimal precision for displayed results.
  7. Press the calculate button.
  8. Review the result above the form.
  9. Export the result as CSV or PDF when needed.

What Is an Infimum?

An infimum is the greatest lower bound of a set. It may be a member of the set. It may also sit outside the set. The idea is useful when a set has values that approach a floor without reaching it.

Why This Calculator Helps

Many learners confuse minimum and infimum. A minimum must belong to the set. An infimum only needs to be the largest number that stays below every set element. This calculator separates those ideas. It accepts finite values and interval unions. It also checks a proposed lower bound.

Working With Data Sets

For a finite list, the infimum equals the smallest entered value. The calculator sorts the values conceptually and reports the lowest one. It also shows whether the value is attained. Finite nonempty data always attain the infimum because the smallest listed number is present.

Working With Intervals

Intervals can be open, closed, bounded, or unbounded. For a union of intervals, the infimum comes from the smallest lower endpoint. If the endpoint is included, the infimum is also a minimum. If the endpoint is open, the set can move close to it without touching it. If an interval extends to negative infinity, the infimum is negative infinity.

Checking a Lower Bound

A lower bound must be less than or equal to every set member. The greatest lower bound is the best possible lower bound. The optional candidate field helps test that idea. If the candidate is greater than the calculated infimum, it fails. If it is equal or smaller, it is a valid lower bound.

Practical Uses

Infimum calculations appear in real analysis, optimization, economics, and constraints. They help describe limits, feasible regions, and best possible guarantees. They also make interval reasoning clearer. Use this tool to verify homework examples, prepare teaching notes, or document quick set checks. Export options support records and sharing. Always review the entered set carefully. A wrong interval endpoint can change the answer.

Good Entry Tips

Use decimals when exact fractions are not needed. Keep each interval logical. The left endpoint must not exceed the right endpoint. Mark open endpoints when the boundary is not included. Recalculate after each change. Save exports only after checking results.

FAQs

What does infimum mean?

The infimum is the greatest lower bound of a set. It is the largest value that remains less than or equal to every set element.

Is the infimum always inside the set?

No. The infimum can be outside the set. For example, the interval (0, 5) has infimum 0, but 0 is not included.

How is infimum different from minimum?

A minimum must belong to the set. An infimum only needs to be the greatest lower bound. Every minimum is an infimum, but not every infimum is a minimum.

Can a finite set have an unattained infimum?

No. A nonempty finite set always contains its smallest value. Therefore, its infimum is attained and equals its minimum.

How do open intervals affect the result?

An open lower endpoint can still be the infimum. It is not a minimum because the endpoint is excluded from the set.

What does negative infinity mean here?

Negative infinity means the set extends downward without a finite lower limit. In that case, no real number can be the greatest lower bound.

Can I check a candidate lower bound?

Yes. Enter a candidate value. The calculator checks whether it is less than or equal to the calculated infimum.

What export formats are available?

You can download the calculated result as a CSV file or PDF file. Both options help save reports, examples, and study notes.

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