Set Builder Calculator
Turn an interval and number rule into set-builder, interval, and roster notation.
Interval details and set rules
Required. Use a standard decimal such as – 2.5 or 10.
Choose whether the left endpoint belongs to the set.
Required. It must be greater than or equal to the left endpoint.
Choose whether the right endpoint belongs to the set.
Select the universe from which members are drawn.
Filter the eligible natural numbers or integers.
Required in this mode. Enter a positive step, such as 0.5 or 2.
Live results
Set-builder notation
{x | 10 < x < 23, x ∈ ℤ, x is odd}
Interval notation
(10, 23)
Cardinality
6
First member
11
Last member
21
Roster form
{11, 13, 15, 17, 19, 21}
Set-builder notation for odd integers between 10 and 23 contains 6 members.
Roster detail
| Index | Element | Membership check |
|---|---|---|
| 1 | 11 | Integer in (10, 23); odd |
| 2 | 13 | Integer in (10, 23); odd |
| 3 | 15 | Integer in (10, 23); odd |
| 4 | 17 | Integer in (10, 23); odd |
| 5 | 19 | Integer in (10, 23); odd |
| 6 | 21 | Integer in (10, 23); odd |
The startup example is fully enumerable, so every member appears in the table and workbook.
How to use the Set Builder Calculator
What this calculator does
This calculator converts a bounded interval and a membership rule into three common mathematical descriptions: set-builder notation, interval notation, and roster form. It also reports the set's cardinality and, when they exist, its first and last members. The tool is designed for finite integer, natural-number, or evenly spaced sets and for bounded real intervals. It does not solve arbitrary symbolic predicates, unions of separate intervals, or function-domain expressions; instead, it creates a precise set from the controls shown above. OpenStax's introduction to basic set concepts and set-builder notation provides useful background on the notation.
When to use it
Use the calculator to check homework involving open or closed intervals, list odd or even integers in a range, identify primes in a bounded interval, create a regularly spaced numerical sequence, or translate a real interval into inequalities. It is also useful for preparing examples for algebra, precalculus, discrete mathematics, and introductory programming lessons where a set must be defined both by a rule and by its members.
How to calculate
- The calculator opens with a ready-to-use demonstration: odd integers strictly between 10 and 23. The results and a validated example XLSX workbook are available immediately.
- Replace the Left endpoint and Right endpoint with standard decimal values. Then set each endpoint rule to Included (closed) or Excluded (open).
- Choose the Type of numbers to consider. Natural numbers and integers reveal the Integer rule; evenly spaced numbers reveal the Spacing field; real numbers need no extra rule.
- Read the live notations, cardinality, boundary members, roster preview, and detail table. Select Download Excel to export the current typed model. Reset clears the demonstration and all semantic data, so Excel export is disabled until a complete valid state is entered again.
Input guide
Left endpoint and Right endpoint are required finite decimals between – 1 trillion and 1 trillion. Examples include 10, – 2.5, and 1,250.75 using a period as the decimal separator. Scientific notation and ambiguous decimal-comma input are rejected. The right endpoint cannot be lower than the left. The two endpoint-rule controls are required selections: “included” means the endpoint itself may belong to the set, while “excluded” means the inequality is strict. Changing either rule can add or remove a boundary member.
Type of numbers to consider is required. Natural numbers means 1, 2, 3, ... in this calculator; zero is not included. Integers permits negative, zero, and positive whole numbers. Evenly spaced numbers starts at the left endpoint when it is included, or one full step after it when excluded. Real numbers represents every real value in the interval and is usually not roster-enumerable.
Integer rule is required for natural numbers and integers. “All eligible numbers” keeps every whole-number member; “even” keeps multiples of 2; “odd” keeps integers not divisible by 2; and “prime” keeps positive integers greater than 1 with no divisors other than 1 and themselves. For a formal prime-number definition, see Wolfram MathWorld's prime number reference. Spacing is required only for evenly spaced numbers and must be a positive finite decimal. A smaller spacing produces more members; a larger spacing produces fewer.
Output guide
Set-builder notation is the exact rule-based description, combining endpoint inequalities, the selected number universe, and any filter. Interval notation records only the interval boundaries; square brackets indicate inclusion and parentheses indicate exclusion. OpenStax's table connecting inequalities, set-builder form, and interval notation explains this correspondence.
Cardinality is the number of members when the set is finite. For a nondegenerate real interval it reads “Uncountably infinite.” First member and Last member identify extrema when they exist; an open real boundary has no least or greatest member. Roster form lists members explicitly when practical. Large finite sets use an abbreviated preview, while real intervals state that a roster is unavailable. The Roster detail table shows the index, element, and membership check for the same canonical members used by the export. A note below the table explains whether all members or a bounded preview is shown.
Worked example
The startup values are left endpoint 10 excluded, right endpoint 23 excluded, number type integers, and integer rule odd. The endpoint rules produce the inequality 10 < x < 23. The first integer above 10 is 11 and the last integer below 23 is 22; applying the odd rule keeps 11, 13, 15, 17, 19, and 21. Therefore the displayed set-builder form is {x | 10 < x < 23, x ∈ ℤ, x is odd}, the roster is {11, 13, 15, 17, 19, 21}, the cardinality is 6, the first member is 11, and the last member is 21. These same values appear in the first-open workbook.
How the enumeration works
For integers, the calculator finds the smallest integer that satisfies the left inequality and the largest integer that satisfies the right inequality. It then applies the selected parity or primality test. Non-integer endpoints are not simply rounded to the nearest whole number: the inequality determines the correct inward integer. For example, an included left endpoint of 2.4 still starts an integer set at 3, while an excluded left endpoint of 2 starts at 3 as well.
For evenly spaced numbers, the left endpoint is the sequence anchor. If that endpoint is included, the first term is the left endpoint; if excluded, the first term is one spacing step later. Terms continue until the right-boundary condition would be violated. Floating-point comparisons use a small scale-aware tolerance, but displayed values are normalized to avoid artifacts such as 0.30000000000000004.
Common interpretation mistakes
- Do not confuse interval notation with roster notation. An interval describes a continuous range unless the number universe adds a discrete restriction.
- Parentheses mean excluded endpoints; square brackets mean included endpoints. Infinity, when used in interval notation elsewhere, always takes a parenthesis.
- Prime numbers are positive integers greater than 1. Negative numbers, zero, and 1 are not prime.
- A finite set may be empty. An empty roster is shown as ∅ and has cardinality 0.
- For large finite sets, the on-page roster and table may be abbreviated for readability, while the workbook includes the complete sequence up to the calculator's safe export limit and clearly labels any preview-only case.