Set Builder Calculator - Interval to Notation and Roster

A set builder calculator turns an interval and a few extra rules into the two standard notations for a set: the set builder form and the roster form. Type the endpoints, pick whether each is included, choose the number type, and the page prints the set builder notation, the inequality, the interval notation, and a roster form so you can read the same set in four ways.

• • Pre-calculus homework: Write a set as { x in Z | 10 < x < 23, x is odd } and read the roster {11, 13, 15, 17, 19, 21} in the same pass.
• • Plotting and counting: Get the explicit list of elements in a finite set so you can count them or check a hand count.
• • Set builder to roster conversion: Convert a set builder expression into a roster form, or write a compact set builder form from a roster list.
• • Real intervals: Generate the set builder form of a real interval such as { x in R | 1 < x < 5 } without trying to list every real number in between.

The set builder calculator is intentionally narrow: it answers the set builder and roster form question for a single interval with one extra condition.

If you only need the bracket form of the same interval, the Interval Notation Calculator page prints the interval, the inequality, and the set builder form for that narrower use case.

The page implements the standard set builder form for a single interval: it adjusts the endpoints to the chosen number type, walks the surviving elements, applies the optional condition, and prints the set builder notation, the roster form, the inequality, and the interval form from the same input.

• a, b: Left and right endpoint values.
• R: Endpoint relation: \u2264 if closed (included), < if open (excluded).
• S: Allowed numbers: Z (integers), N (naturals), evenly-spaced values, or R (all reals).
• condition: Optional rule: even, odd, prime, or no rule.
• roster form: Same set written as a comma-separated list inside curly braces.

The same pattern works for every other number type. Naturals start at 1, evenly-spaced values use the spacing as the step, and reals return the set builder form with a note that the roster is uncountable. When the endpoints are decimals but the number type is integers, the page rounds each endpoint outward to the nearest in-set integer.

According to Khan Academy, set builder notation describes a set as the collection of all elements x such that x satisfies a stated condition, usually written as {x | condition}.

When the problem is written as a single-variable inequality like 10 < x < 23 and you want to solve it for x or graph it on a number line, the Linear Inequality Calculator page handles that side of the same problem.

Four ideas explain what a set builder expression really says and how the page turns it into a roster form.

These four ideas cover every supported input. The endpoints pick the interval, the endpoint types pick the bracket style, the number type picks the domain, and the condition narrows the set. When the domain is R and the interval is non-empty, the roster form is uncountable, so the page falls back to the set builder form for the real number type.

If you want to see the same set drawn on a number line with filled or open dots at the endpoints, the Graphing Inequalities 1D Calculator page plots the interval and marks the bracket style visually.

Five short steps cover every common case, from a clean textbook example to a real-number interval.

1. 1 Enter the two endpoints: Type the left and right values of the interval. The default example is 10 and 23, so the page opens on a clean odd-integer case.
2. 2 Pick open or closed at each end: Choose whether each endpoint is included (closed) or excluded (open). The page turns those choices into the bracket symbols, the inequality signs, and the page logic for adjusting decimal endpoints.
3. 3 Choose the type of numbers: Pick integers, natural numbers, evenly-spaced values, or reals. Integers allow negative values, natural numbers start at 1, evenly-spaced values use the spacing field, and reals skip the roster form.
4. 4 Add an extra condition if needed: Pick even, odd, prime, or no extra condition. Even, odd, and prime only work for integers or natural numbers; the page falls back to no condition otherwise.
5. 5 Set the spacing for evenly-spaced values: If you picked evenly-spaced values, type the step. The page uses it as the stride when walking the interval; spacing must be positive.

Try endpoints 5 and 20, both closed, with the integer number type and the prime condition. The page gives the set builder form { x in Z | 5 \u2264 x \u2264 20, x is prime }, the roster form {5, 7, 11, 13, 17, 19}, the inequality 5 \u2264 x \u2264 20, the interval [5, 20], and the element count 6.

When the extra condition is prime and you want to confirm a single value rather than list every prime in the interval, the Prime Number Checker page tests one number at a time and explains why it is or is not prime.

These benefits matter most when you need both notations to match.

• • See the set in four notations at once: The page prints the set builder form, the roster form, the inequality, and the interval form from the same input.
• • Skip the off-by-one errors on endpoints: The page applies the endpoint type to the inequality, the bracket symbol, and the candidate walk, so the four notations always agree.
• Count and list the elements automatically: For finite sets, the page prints the roster list and the count.
• • Switch between integers, naturals, evenly-spaced, and reals: The four supported number types cover the most common pre-calculus cases.
• • Handle real intervals without breaking: Real intervals have uncountably many elements, so the page falls back to the set builder form and a short notice.

The page is most useful as a notation and counting aid, not as a replacement for understanding set builder form. Use it to confirm that the set builder expression, the roster form, and the interval all refer to the same set.

When the number type is evenly-spaced, the set is just an arithmetic sequence with a known first term and common difference, and the Arithmetic Sequence Calculator page gives the nth term, the partial sums, and the same step-based enumeration.

The four inputs that drive the page are the endpoints, the endpoint types, the number type, and the optional condition. Each one shifts the result in a predictable way.

• • The page only handles a single interval with a single extra condition. It does not compute unions, intersections, or complements of multiple intervals.
• • For real intervals, the roster form is intentionally omitted. The page returns the set builder form and a notice that the roster would be uncountable.
• • The prime filter uses a trial-division test up to the square root, which is correct for the small ranges a homework problem uses.

According to Wolfram MathWorld, a set is a finite or infinite collection of objects, and the set builder form {x | P(x)} is read as the set of all x for which the predicate P(x) holds.

When the set builder expression you are trying to write comes from solving an absolute value inequality such as |x - 5| < 3, the Absolute Value Inequality Calculator page solves the inequality and returns the matching interval, which you can then paste back into this page to format it as a set builder expression.