Converter inputs
Example data table
| Input | Interval | Inequality | Set-builder |
|---|---|---|---|
| {1,2,3,4,5} | [1, 5] | 1 ≤ x and x ≤ 5 | { x ∈ ℝ | 1 ≤ x and x ≤ 5 } |
| (0, 5] | (0, 5] | 0 < x and x ≤ 5 | { x ∈ ℝ | 0 < x and x ≤ 5 } |
| 2 < x <= 7 | (2, 7] | 2 < x and x ≤ 7 | { x ∈ ℝ | 2 < x and x ≤ 7 } |
| (0, 5] ∪ [8, 10) | (0, 5] ∪ [8, 10) | 0 < x and x ≤ 5 OR 8 ≤ x and x < 10 | { x ∈ ℝ | 0 < x and x ≤ 5 } ∪ { x ∈ ℝ | 8 ≤ x and x < 10 } |
Table values are illustrative; your results depend on input rules.
Formula used
- [a, b] means a ≤ x ≤ b.
- (a, b) means a < x < b.
- (a, b] means a < x ≤ b.
- [a, b) means a ≤ x < b.
- Union combines: A ∪ B means “in A or in B”.
- Set-builder form: { x ∈ ℝ | condition }.
How to use this calculator
- Enter your set in roster, interval, or inequality format.
- Pick “Auto detect” unless you want strict parsing.
- Choose the variable and domain for set-builder output.
- Set a sampling step if you want roster examples.
- Press Submit to see results above the form.
- Use Download CSV or PDF for quick reporting.
Why conversions matter in proofs and problem solving
Set descriptions appear in different forms across algebra, calculus, and discrete mathematics. Converting between roster, interval, inequality, and set-builder notation reduces ambiguity and helps you apply the correct theorem conditions quickly.
Endpoint rules and accuracy checks
Brackets indicate inclusion, while parentheses indicate exclusion. This tool maps those endpoint rules directly into inequalities. For example, (2, 7] becomes 2 < x ≤ 7, preserving strictness on the left and inclusion on the right.
Union handling for disconnected solution sets
Many real problems produce multiple valid ranges, such as absolute value inequalities or rational constraints. Union notation combines these ranges. The converter supports expressions like (0, 5] ∪ [8, 10) and returns an equivalent “OR” inequality statement for clear reasoning.
Roster patterns and when intervals are justified
A roster list is finite and exact. When listed elements are consecutive integers, the converter can represent them as a closed interval. If the list follows an arithmetic progression, it expresses a parameter form in set-builder notation using an integer index.
Sampling and visualization for interpretation
For finite intervals, the roster output is generated by sampling using your chosen step size. The Plotly graph then displays each interval as a segment on a number line and overlays sample points. This makes it easy to spot gaps, overlaps, and endpoint inclusions visually.
Export-ready outputs for study and documentation
CSV export provides a compact record of the converted fields for notes or spreadsheets. PDF export produces a printable summary with input, detected type, conversions, and remarks. These exports support consistent formatting when sharing solutions with classmates or instructors.
FAQs
1) What inputs does the converter accept?
It accepts roster lists like {1,2,3}, interval ranges like (0,5], unions using ∪, and inequalities such as 2 < x ≤ 7. It also understands compact ranges like 1..5 inside rosters.
2) Why does a roster sometimes not become a single interval?
A finite roster can represent many different sets. Only consecutive integers uniquely match a closed interval. Otherwise, the safest output keeps roster form and provides set-builder patterns when a clear arithmetic structure is detected.
3) How are open and closed endpoints handled?
Brackets mean the endpoint is included, so the inequality uses ≤. Parentheses mean exclusion, so the inequality uses <. The converter applies the same rule consistently for both left and right endpoints.
4) Can I use infinity in intervals or inequalities?
Yes. Use inf, +inf, or -inf. For example, [3, inf) corresponds to x ≥ 3. Roster sampling is limited for infinite bounds, but the interval, inequality, and set-builder outputs remain valid.
5) What does “Auto detect” do?
Auto detect tries interval notation first, then inequalities, then roster lists. If your expression could match multiple styles, selecting an explicit input type forces stricter parsing and reduces surprises.
6) How should I read the Plotly graph?
Each interval is drawn as a horizontal segment. Filled endpoints indicate inclusion; open endpoints indicate exclusion. Sample points show example members for finite ranges, helping you verify that the visual matches the converted inequality and set-builder forms.