Union and Intersection of Intervals Aleks Calculator

This calculator helps you find the union and intersection of intervals. Whether you're working with real numbers, integers, or other mathematical sets, this tool provides clear results and visualizations to understand how intervals overlap and combine.

What Are Intervals?

An interval is a set of real numbers between two endpoints. Intervals are often represented using interval notation, which uses square brackets [ ] for closed intervals (including endpoints) and parentheses ( ) for open intervals (excluding endpoints).

For example:

[a, b] represents all numbers x such that a ≤ x ≤ b
(a, b) represents all numbers x such that a < x < b
[a, b) represents all numbers x such that a ≤ x < b
(a, b] represents all numbers x such that a < x ≤ b

Intervals can be finite or infinite. For example, [3, ∞) represents all numbers greater than or equal to 3, and (-∞, 5] represents all numbers less than or equal to 5.

Union of Intervals

The union of two intervals is the set of all numbers that are in either interval. In interval notation, the union is represented by the ∪ symbol.

Formula: A ∪ B = {x | x ∈ A or x ∈ B}

When finding the union of two intervals, you combine them into the smallest interval that contains both. If the intervals overlap or are adjacent, they merge into a single interval.

For example:

[1, 3] ∪ [2, 4] = [1, 4]
(1, 3) ∪ (2, 4) = (1, 4)
[1, 3] ∪ [4, 5] = [1, 3] ∪ [4, 5]

Intersection of Intervals

The intersection of two intervals is the set of all numbers that are in both intervals. In interval notation, the intersection is represented by the ∩ symbol.

Formula: A ∩ B = {x | x ∈ A and x ∈ B}

When finding the intersection of two intervals, you look for the overlapping portion. If there is no overlap, the intersection is the empty set, denoted by ∅.

For example:

[1, 3] ∩ [2, 4] = [2, 3]
(1, 3) ∩ (2, 4) = (2, 3)
[1, 3] ∩ [4, 5] = ∅

Examples

Example 1: Union of Intervals

Find the union of [1, 5] and [3, 7].

The intervals overlap from 3 to 5, so the union is [1, 7].

Example 2: Intersection of Intervals

Find the intersection of (2, 6) and [4, 8].

The overlapping portion is [4, 6), so the intersection is [4, 6).

Example 3: No Intersection

Find the intersection of [1, 3] and [4, 5].

There is no overlap, so the intersection is ∅.

FAQ

What is the difference between union and intersection?

The union of two sets includes all elements that are in either set, while the intersection includes only elements that are in both sets.

How do I represent an empty set in interval notation?

An empty set is represented by ∅. It occurs when two intervals have no overlap.

Can intervals be infinite?

Yes, intervals can be infinite. For example, [3, ∞) represents all numbers greater than or equal to 3, and (-∞, 5] represents all numbers less than or equal to 5.

What is the difference between open and closed intervals?

A closed interval includes its endpoints (using square brackets), while an open interval excludes its endpoints (using parentheses).