Union and Intersection Intervals Calculator

Intervals are ranges of numbers between two endpoints. Calculating the union and intersection of intervals helps in solving problems in mathematics, physics, engineering, and computer science. 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 but not the other (e.g., [a, b) or (a, b])

Intervals are commonly used in calculus, real analysis, and optimization problems. Understanding how to combine and compare intervals is essential for solving many mathematical and practical problems.

Union of Intervals

The union of two intervals is the set of all numbers that are in either interval. The union of intervals [a, b] and [c, d] is:

[a, b] ∪ [c, d] = [min(a, c), max(b, d)]

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

If the intervals do not overlap, the union is simply the combination of both intervals. If they overlap, the union is the smallest interval that contains both original intervals.

Intersection of Intervals

The intersection of two intervals is the set of all numbers that are in both intervals. The intersection of intervals [a, b] and [c, d] is:

[a, b] ∩ [c, d] = [max(a, c), min(b, d)]

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

If the intervals do not overlap, the intersection is empty. If they overlap, the intersection is the largest interval that is contained within both original intervals.

How to Use This Calculator

Enter the first interval in the format [a, b], (a, b], [a, b), or (a, b).
Enter the second interval using the same format.
Click "Calculate" to find the union and intersection.
Review the results and chart visualization.

The calculator will display the union and intersection of the intervals, along with a visual representation of the intervals on a number line.

FAQ

What is the difference between union and intersection of intervals?

The union of intervals includes all numbers from either interval, while the intersection includes only numbers that are in both intervals.

How do I represent an open interval?

Use parentheses for open intervals, such as (a, b) for an open interval between a and b.

What if the intervals don't overlap?

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