How this calculator works
Set builder notation names a set by stating which numbers belong to it. The pattern is a variable, the number system it comes from, and a rule separated by a vertical bar. Roster form instead lists every member between braces. An interval of real numbers is infinite, so it has no roster form; whole numbers and natural numbers are discrete, so their intervals can be listed. Filters such as even, odd or prime remove some members, and evenly spaced numbers form an arithmetic progression with a chosen gap.
Formula
{ x ∈ S | condition } means the set of every element x of S that satisfies the condition. For an interval, the condition is left ≤ x ≤ right, with < in place of ≤ where an endpoint is excluded.
Worked example
The integers from 0 to 10 inclusive are written { x ∈ ℤ | 0 ≤ x ≤ 10 } in set builder notation and {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} in roster form. Adding the filter x is even leaves {0, 2, 4, 6, 8, 10}.
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Frequently asked questions
What is the difference between roster and set builder notation?
Roster form lists the members explicitly, which works only for finite sets or sets short enough to list. Set builder notation describes a rule, so it works for infinite sets such as all real numbers between 0 and 1.
Do the natural numbers include zero?
There are two conventions. This calculator uses the one that includes zero, so ℕ = {0, 1, 2, ...}. If your course starts the natural numbers at 1, set the left endpoint to 1 and the two answers agree.
What happens when an endpoint is excluded?
The endpoint is not a member of the set, so it is dropped from the roster and the set builder notation uses a strict inequality at that end. In the evenly spaced case, the progression still starts at the endpoint value but that value is skipped.
Why is the roster cut off at 100 values?
A page cannot list an unbounded number of members. When more than 100 values qualify, the first 100 are shown with an ellipsis. Narrow the interval or add a filter to see the rest.