Set Builder and Interval Notation Calculator

Set builder notation and interval notation are two common ways to represent sets of numbers in mathematics. This calculator helps you convert between these two notations and understand their differences.

What is Set Builder Notation?

Set builder notation is a method of describing a set by specifying the properties that its members must satisfy. It's written in the form:

{x | x ∈ S, P(x)}

Where:

x is the variable representing elements of the set
S is the universal set from which elements are drawn
P(x) is the property that elements must satisfy

For example, the set of all positive even integers less than 10 can be written as:

{x | x ∈ ℤ, x > 0, x < 10, x is even}

What is Interval Notation?

Interval notation is a way to represent a set of real numbers that lie between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included or excluded.

Note: Interval notation is typically used for real numbers, not integers or other number systems.

The main symbols used in interval notation are:

( ) - Parentheses indicate that an endpoint is not included
[ ] - Brackets indicate that an endpoint is included
∞ - Infinity symbol is used for unbounded intervals

For example, the interval from 2 to 5, including both endpoints, is written as [2, 5].

Converting Between Notations

Converting between set builder notation and interval notation requires understanding the properties of the set and the range of numbers involved.

From Set Builder to Interval Notation

Identify the universal set (usually ℝ for real numbers)
Determine the range of values that satisfy the property P(x)
Identify whether the endpoints are included or excluded
Write the interval using appropriate brackets or parentheses

From Interval to Set Builder Notation

Identify the interval endpoints and whether they're included or excluded
Express the inclusion/exclusion as inequalities
Write the set builder notation with the appropriate property

Examples

Example 1: Positive real numbers

Set builder notation: {x | x ∈ ℝ, x > 0}

Interval notation: (0, ∞)

Example 2: Integers between -3 and 3, inclusive

Set builder notation: {x | x ∈ ℤ, -3 ≤ x ≤ 3}

Interval notation: [-3, 3]

Example 3: Real numbers greater than 5 but less than or equal to 10

Set builder notation: {x | x ∈ ℝ, 5 < x ≤ 10}

Interval notation: (5, 10]

FAQ

Can I use interval notation for integers?: Technically yes, but interval notation is most commonly used for real numbers. For integers, set builder notation is often more precise.
What's the difference between [ ] and ( ) in interval notation?: Brackets [ ] indicate that the endpoint is included in the set, while parentheses ( ) indicate it's excluded.
Can I use interval notation for complex numbers?: No, interval notation is specifically for real numbers. For complex numbers, set builder notation is required.
How do I represent an empty set in interval notation?: An empty set is represented by two parentheses or brackets with the same number on both sides, like (5, 5) or [3, 3].
Can I use interval notation for sets of functions?: No, interval notation is only for sets of real numbers. For function sets, set builder notation is needed.