Turn a list, interval or condition into proper set-builder notation — with roster form and a plain reading.
Set-builder notation describes a set by a rule, not by listing every member: {x | rule}. Intervals describe real numbers; conditions over a range list the integers that satisfy the rule.
Set-builder notation
How to use this tool
A list of numbers, an interval, or a condition over a range.
The list, the interval ends, or the range and condition.
The big result is the rule-based notation.
Roster form lists the members — handy for checking.
Formula & worked example
The list 2, 4, 6, 8, 10 is the arithmetic set {x | x = 2 + 2n, n ∈ ℕ, 2 ≤ x ≤ 10}, written in roster form as {2, 4, 6, 8, 10}. The interval [0, 10] is {x ∈ ℝ | 0 ≤ x ≤ 10}.
Limitations — read this before you rely on it
A finite list that is not arithmetic has no shorter rule, so its set-builder form is just its roster form. Intervals contain infinitely many real numbers, so they cannot be listed. Prime and condition sets are computed over the integer range you enter, not over all integers.
Frequently asked questions
A way to describe a set by a rule instead of listing members: {x | x is even} means the set of all x such that x is even. The vertical bar | reads as “such that”.
∈ means “is an element of”. ℝ is all real numbers, ℤ is all integers (…, −2, −1, 0, 1, 2, …), and ℕ is the natural numbers (1, 2, 3, …).
As {x ∈ ℝ | 0 ≤ x ≤ 10}. For an open interval (0, 10), use strict inequalities: {x ∈ ℝ | 0 < x < 10}.
As {x ∈ ℤ | x is even, 0 ≤ x ≤ 10}, which in roster form is {0, 2, 4, 6, 8, 10}.
Listing the members inside braces, like {2, 4, 6, 8, 10}. It works for finite sets but not for infinite ones like all real numbers in an interval.