Set Builder Notation Calculator
Convert inequalities, intervals, or lists into set-builder, roster, and interval forms; choose domain and element properties for accurate results quickly.
Inputs
Actions
Equation Preview
Helping Notes
Notes: Use interval notation (e.g. [2,8) ), inequalities (e.g. 3 ≤ x ≤ 7), or comma lists (e.g. 1,2,3,4). Enumeration works for integer domains.
If endpoints are non-integer or domain is Q/R, roster form may not be listed; the calculator will show set-builder and interval representations instead.
Results
Roster notation:
Set-builder notation:
Interval notation:
Steps / Notes:
What is Set Builder Notation Calculator
A Set Builder Notation Calculator is a tool that interprets compact set definitions written in set-builder form and converts them into explicit roster lists, human-readable descriptions, or predicate forms. Set-builder notation expresses sets by a rule or property rather than by listing all elements directly. This calculator helps learners and practitioners check their understanding, verify membership, and transform between representations for work in discrete math, logic, database queries, or programming.
About the Set Builder Notation Calculator
This calculator supports common set-builder forms such as membership constraints, arithmetic generators, and interval-style definitions. It accepts expressions like \( \{ x \in S \mid P(x) \} \), generator patterns such as \( \{ f(n) \mid n \in \mathbb{Z} \} \), and inequality-based sets like \( \{ x \in \mathbb{R} \mid a \le x \le b \} \). The tool is useful for verifying finite sets, previewing the first N generated elements for infinite definitions, and producing a predicate or plain-English translation. The formulas are provided in render-ready math markup so they display cleanly with a math renderer on the page.
How to Use this Set Builder Notation Calculator
1. Enter the set-builder expression in standard form, for example: \( \{ x \in \mathbb{N} \mid x < 6 \} \) or \( \{ 2n \mid n \in \mathbb{Z} \} \).
2. Choose whether you want a finite roster output (explicit list), a predicate translation, or a generator preview (first N elements for infinite sets).
3. Click Calculate. For finite sets the calculator lists members explicitly. For generator-based or infinite sets it shows a sampled sequence plus the rule.
4. Use the translator output to verify membership questions (e.g., is 7 in the set?) or to export the roster for examples and exercises.
Formulas (render-ready)
General set-builder structure:
\( \; A = \{ x \mid P(x) \} \; \)
Membership with domain:
\( \; A = \{ x \in S \mid P(x) \} \; \)
Generator form:
\( \; B = \{ f(n) \mid n \in D \} \; \)
Examples of Set Builder Notation Calculator
Example 1 — Finite natural numbers:
\( \{ n \in \mathbb{N} \mid n \le 5 \} \Rightarrow \{0,1,2,3,4,5\} \).
Example 2 — Even integers:
\( \{ 2k \mid k \in \mathbb{Z} \} \Rightarrow \{ \dots, -4, -2, 0, 2, 4, \dots \} \) and the calculator can preview \( \{-6,-4,-2,0,2,4,6\} \).
Example 3 — Interval expressed as set-builder:
\( \{ x \in \mathbb{R} \mid -1 \le x < 3 \} \Rightarrow \) interval representation and a predicate translation "all real x where -1 ≤ x < 3".
Responsive formula notes
Formulas above use math markup that will scale responsively when rendered by a math rendering library on the page; keep them in inline or display mode so they reflow on narrow screens and remain readable across devices.
What does set builder notation mean?
Set builder notation defines a set by stating a property or rule that its members satisfy rather than listing every element explicitly.
How do I convert set builder notation to a list?
Apply the defining rule to generate elements; if finite, list them all. If infinite, provide a generator pattern or preview a finite sample.
Can the calculator handle infinite sets?
Yes — it can show generator patterns and preview the first N elements, but it cannot list infinitely many elements in full.
What input formats are supported?
Common forms include \( \{ x \mid P(x) \} \), \( \{ x \in S \mid P(x) \} \), and generator forms like \( \{ f(n) \mid n \in D \} \).
Is set-builder notation only for numbers?
No — it can describe elements from any domain (numbers, ordered pairs, strings) using an appropriate predicate or domain specification.
How do I check if an element belongs to a set?
Substitute the element into the predicate P(x); if the predicate is true within the domain, the element is in the set.
Can I express intervals with set-builder notation?
Yes — intervals translate naturally, e.g., \( \{ x \in \mathbb{R} \mid a \le x < b \} \) represents a half-open interval.
Why use set-builder notation?
It offers concise, flexible descriptions for complex or infinite sets and clarifies membership rules used in proofs and programming logic.