Union and Intersection of Interval Notation Calculator

Interval notation is a concise way to represent sets of real numbers. This calculator helps you find the union and intersection of two intervals, which are fundamental operations in set theory and algebra.

What is Interval Notation?

Interval notation is a method of representing a set of real numbers that lie between two endpoints. It's commonly used in mathematics, engineering, and science to describe ranges of values.

There are four types of intervals:

Closed interval: Includes both endpoints (e.g., [a, b])
Open interval: Excludes both endpoints (e.g., (a, b))
Half-open interval: Includes one endpoint and excludes the other (e.g., [a, b) or (a, b])
Infinite interval: Extends to infinity (e.g., [a, ∞) or (-∞, b])

Note: The square brackets [ ] indicate that the endpoint is included, while parentheses ( ) indicate that the endpoint is excluded.

Union of Intervals

The union of two intervals A and B, denoted A ∪ B, is the set of all elements that are in A, in B, or in both. In interval notation, the union is represented by combining the intervals when they overlap or are adjacent.

If A = [a, b] and B = [c, d], then A ∪ B = [min(a, c), max(b, d)] when the intervals overlap or are adjacent.

For example, [1, 5] ∪ [3, 7] = [1, 7].

Intersection of Intervals

The intersection of two intervals A and B, denoted A ∩ B, is the set of all elements that are in both A and B. In interval notation, the intersection is represented by the overlapping portion of the two intervals.

If A = [a, b] and B = [c, d], then A ∩ B = [max(a, c), min(b, d)] when the intervals overlap.

For example, [1, 5] ∩ [3, 7] = [3, 5]. If the intervals do not overlap, the intersection is the empty set ∅.

Examples

Let's look at some examples of union and intersection operations:

Example 1: Overlapping Intervals

Interval A: [2, 6]

Interval B: [4, 8]

Union: [2, 8]
Intersection: [4, 6]

Example 2: Adjacent Intervals

Interval A: [1, 3]

Interval B: [3, 5]

Union: [1, 5]
Intersection: {3}

Example 3: Non-overlapping Intervals

Interval A: [1, 3]

Interval B: [5, 7]

Union: [1, 3] ∪ [5, 7]
Intersection: ∅ (empty set)

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 ∅ or sometimes as an interval that doesn't contain any numbers, like [a, b] where a > b.

Can I use this calculator for infinite intervals?

Yes, you can use ∞ or -∞ in the calculator to represent infinite intervals. For example, [5, ∞) represents all numbers greater than or equal to 5.

What if my intervals don't overlap?

If the intervals don't overlap, the intersection will be the empty set ∅, and the union will be the combination of both intervals.