Union and Intersections of Intervals Calculator

Intervals are ranges of numbers between two endpoints. Calculating the union and intersection of intervals helps in solving problems in mathematics, computer science, and engineering. This calculator helps you find the combined range (union) and overlapping range (intersection) of two intervals.

What Are Intervals?

An interval is a set of real numbers between two endpoints. Intervals can be open, closed, or half-open:

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])

Intervals are fundamental in mathematics and computer science for representing ranges of values.

Union of Intervals

The union of two intervals is the smallest interval that contains all elements from both intervals. It combines the ranges of both intervals.

Formula: Union of [a, b] and [c, d] is [min(a, c), max(b, d)]

For example, the union of [1, 5] and [3, 7] is [1, 7].

Intersection of Intervals

The intersection of two intervals is the set of all elements that are common to both intervals. It represents the overlapping range between the two intervals.

Formula: Intersection of [a, b] and [c, d] is [max(a, c), min(b, d)] if max(a, c) ≤ min(b, d), otherwise there is no intersection.

For example, the intersection of [1, 5] and [3, 7] is [3, 5].

How to Use This Calculator

Enter the lower and upper bounds of the first interval.
Select whether the interval is open, closed, or half-open.
Enter the lower and upper bounds of the second interval.
Select the interval type for the second interval.
Click "Calculate" to see the union and intersection results.

The calculator will display the combined range (union) and overlapping range (intersection) of the two intervals.

Frequently Asked Questions

What is the difference between union and intersection of intervals?: The union combines all elements from both intervals, while the intersection contains only the elements common to both intervals.
Can intervals overlap?: Yes, intervals can overlap. The intersection will be the overlapping range if it exists.
What happens if intervals do not overlap?: If intervals do not overlap, there is no intersection. The union will be the combined range of both intervals.
How are open and closed intervals different?: Closed intervals include their endpoints, while open intervals exclude them. Half-open intervals include one endpoint and exclude the other.