Union of Two Intervals Calculator

This calculator helps you find the union of two intervals on the real number line. The union of two intervals is the set of all numbers that belong to either interval. This concept is fundamental in mathematics, particularly in set theory and real analysis.

What is the Union of Two Intervals?

In mathematics, an interval is a set of real numbers that lie between two endpoints. The union of two intervals combines all numbers that are in either interval. There are three possible outcomes when taking the union of two intervals:

Disjoint intervals: The intervals do not overlap. The union is simply the combination of both intervals.
Overlapping intervals: The intervals share some numbers. The union covers the entire range from the smallest lower bound to the largest upper bound.
Contained intervals: One interval is entirely within the other. The union is just the larger interval.

Note: The union of two intervals is always an interval itself, unless the intervals are disjoint and not connected by any other interval.

How to Calculate the Union of Two Intervals

To find the union of two intervals [a, b] and [c, d], follow these steps:

Identify the smallest lower bound: min(a, c)
Identify the largest upper bound: max(b, d)
If the intervals overlap or are adjacent, the union is [min(a, c), max(b, d)]
If the intervals are disjoint and not adjacent, the union is the combination of both intervals

Formula: Union of [a, b] and [c, d] = [min(a, c), max(b, d)] if the intervals overlap or are adjacent

The calculator below implements this logic automatically. Simply enter the endpoints of your two intervals and click "Calculate".

Examples of Union Calculations

Let's look at some examples to understand how the union of two intervals works:

Example 1: Overlapping Intervals

Interval 1: [2, 6]
Interval 2: [4, 8]

The union is [2, 8] because 4 is between 2 and 6, and 6 is between 4 and 8.

Example 2: Disjoint Intervals

Interval 1: [1, 3]
Interval 2: [5, 7]

The union is [1, 3] ∪ [5, 7] because there are no numbers common to both intervals.

Example 3: Contained Intervals

Interval 1: [0, 10]
Interval 2: [3, 7]

The union is [0, 10] because [3, 7] is entirely within [0, 10].

Example 4: Adjacent Intervals

Interval 1: [1, 4]
Interval 2: [4, 6]

The union is [1, 6] because the intervals touch at 4.

FAQ

What is the difference between union and intersection of intervals?

The union of two intervals includes all numbers that are in either interval, while the intersection includes only numbers that are in both intervals. For example, the union of [1, 3] and [2, 4] is [1, 4], while the intersection is [2, 3].

Can the union of two intervals be empty?

No, the union of two intervals cannot be empty because intervals are defined as sets of real numbers between two endpoints. Even if the intervals are disjoint, their union will still contain all numbers from both intervals.

Is the union of two intervals always an interval?

Yes, the union of two intervals is always an interval, unless the intervals are disjoint and not connected by any other interval. In that case, the union is the combination of two separate intervals.

How is the union of intervals used in real-world applications?

The concept of interval union is used in various fields such as computer science, engineering, and economics. For example, in scheduling problems, the union of time intervals helps determine the total time covered by multiple events.