Set Builder Notation Calculator

Enter your interval endpoints and choose your interval type (open, closed, or half-open) to get the corresponding set builder notation, interval notation, and roster form. Select the number domain (ℝ, ℤ, ℕ) and an optional element condition (all, odd, even, prime) to filter members. The calculator outputs the set builder expression, the full interval notation string, and a roster list of matching elements.

The starting value of your interval.

The ending value of your interval.

Choose the set of numbers to draw elements from.

Optionally filter which elements qualify within the interval.

Results

Set Builder Notation

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Interval Notation

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Number of Elements in Roster

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Roster Form

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Element Distribution in Roster

Results Table

Frequently Asked Questions

What is set builder notation?

Set builder notation is a mathematical shorthand for describing a set by specifying a rule or condition that its elements must satisfy. For example, {x ∈ ℤ | 1 ≤ x ≤ 10} means 'all integers x such that x is between 1 and 10, inclusive.' It is widely used in algebra, calculus, and logic to express sets compactly.

What is the difference between open and closed intervals?

A closed interval [a, b] includes both endpoints a and b, while an open interval (a, b) excludes them. A half-open interval like [a, b) includes a but not b. In set builder notation, closed endpoints use ≤ and open endpoints use <.

How do I represent the set builder form for the odd numbers in [5, 15)?

Set the lower bound to 5 (closed) and the upper bound to 15 (open), choose domain ℤ, and select condition 'odd'. The resulting set builder notation is {x ∈ ℤ | 5 ≤ x < 15, x is odd}, which yields the roster {5, 7, 9, 11, 13}.

What is the roster form of a set?

The roster form (also called the tabular form) lists all elements of a set explicitly, separated by commas and enclosed in curly braces — for example, {2, 4, 6, 8}. It is the most readable form when the set is finite and small, as opposed to set builder notation which uses a rule.

How do you write {5, 10, 15, 20, 25} in set builder notation?

You can write this as {x ∈ ℕ | x = 5k, k ∈ ℕ, 1 ≤ k ≤ 5}, or more simply as {x ∈ ℕ | 5 ≤ x ≤ 25, x is a multiple of 5}. The key is to identify the rule that all listed elements share.

What domains (ℕ, ℤ, ℚ, ℝ) can I use in set builder notation?

ℕ refers to natural numbers (positive integers starting from 1), ℤ includes all integers (negative, zero, positive), ℚ covers rational numbers (fractions), and ℝ covers all real numbers including irrationals. The domain you choose defines which numbers are eligible to be tested against your interval condition.

How does interval notation relate to set builder notation?

Interval notation is a compact way to write a continuous range using brackets and parentheses, such as [1, 10) or (−∞, 5]. Set builder notation is more expressive — it can add domain and condition restrictions on top of the interval. Both describe the same underlying set, but set builder notation lets you filter by properties like odd, even, or prime.

Can this calculator handle prime number filtering?

Yes. Select 'Prime numbers only' in the Element Condition dropdown. The calculator will list all prime numbers that fall within your specified interval and domain. For example, in [1, 20] over ℤ with prime condition, the roster would be {2, 3, 5, 7, 11, 13, 17, 19}.

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