# Set Builder Calculator

URL: https://speedycalculator.com/calculators/math/set-builder/
Last updated: 2026-10-30

> The set builder calculator writes a set of numbers between two endpoints – natural numbers, integers, evenly spaced rationals or reals, optionally even, odd or prime – in both set-builder and roster notation.

## Inputs

- Left endpoint
- Right endpoint
- Endpoints
- Type of numbers
- Condition
- Spacing

## Results

- Set-builder notation
- Roster notation
- Number of members

## Formula

```
{x | a ≤ x ≤ b, x ∈ ℤ, condition}  ↔  {listed members}
```

## Explanation

Set-builder notation describes a set by a rule ("all even integers between 1 and 20"), while roster notation lists its members. Moving between them is a key skill for algebra, probability and computer science.

### How to use the Set Builder Calculator

- **Left endpoint**
- **Right endpoint**
- **Endpoints**
- **Type of numbers**
- **Condition**
- **Spacing**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Choose the endpoints, whether they are included, the kind of numbers and an optional condition. The calculator lists the members (up to 200) and writes the rule.

## Worked example

Inputs — Left endpoint: 1; Right endpoint: 20; Endpoints: Both inclusive; Type of numbers: Integers ℤ; Condition: Prime numbers.

Results — **Set-builder notation = {x | 1 ≤ x ≤ 20, x ∈ ℤ, x is prime}**; Roster notation = {2, 3, 5, 7, 11, 13, 17, 19}; Number of members = 8.

## Key terms

- **Set-builder notation**: {x | rule}, the set of x satisfying a rule.

## FAQ

**What does {x**
x > 3} mean? | The set of all x such that x is greater than 3.

**What is roster notation?**
Listing the members, e.g. {2, 3, 5, 7}.

**Why can't real intervals be listed?**
They contain uncountably many numbers.

**Does ℕ include 0?**
Conventions differ; this calculator treats ℕ as the non-negative integers in the range.


## Sources

- [OpenStax – Contemporary Mathematics](https://openstax.org/details/books/contemporary-mathematics)
- [Wolfram MathWorld – Set-builder notation](https://mathworld.wolfram.com/SetBuilderNotation.html)