Interval Notation Union Calculator

Interval notation is a concise way to represent sets of real numbers. The union of two intervals combines all numbers that belong to either interval. This calculator helps you find the union of two intervals in interval notation.

What is Interval Notation?

Interval notation is a mathematical shorthand used to describe ranges of real numbers. It's commonly used in calculus, algebra, and analysis. The basic forms are:

(a, b) - Open interval: includes all numbers greater than a and less than b
[a, b] - Closed interval: includes all numbers greater than or equal to a and less than or equal to b
(a, b] - Half-open interval: includes all numbers greater than a and less than or equal to b
[a, b) - Half-open interval: includes all numbers greater than or equal to a and less than b

Interval notation is particularly useful when dealing with continuous functions and limits. It provides a compact way to represent infinite sets of numbers.

How to Find the Union of Intervals

The union of two intervals A and B, denoted as A ∪ B, is the set of all elements that are in A, in B, or in both. To find the union of two intervals in interval notation:

Identify the lower bound as the smaller of the two starting points
Identify the upper bound as the larger of the two ending points
Determine the correct bracket type based on the original intervals

Formula for Interval Union

If A = [a₁, a₂] and B = [b₁, b₂], then A ∪ B = [min(a₁, b₁), max(a₂, b₂)]

When the intervals are not adjacent or overlapping, the union is simply the combination of both intervals. If they overlap or are adjacent, the union becomes a single continuous interval.

Examples of Interval Union

Example 1: Overlapping Intervals

Find the union of [2, 5] and [4, 8].

The lower bound is min(2, 4) = 2, and the upper bound is max(5, 8) = 8. The result is [2, 8].

Example 2: Adjacent Intervals

Find the union of [1, 3] and [3, 6].

The intervals are adjacent at 3, so the union is [1, 6].

Example 3: Disjoint Intervals

Find the union of [0, 2] and [5, 7].

Since the intervals don't overlap or touch, the union is [0, 2] ∪ [5, 7].

Note: The union of intervals is always a set of intervals, even if it's just one continuous interval.

FAQ

What is the difference between union and intersection of intervals?

The union of two intervals includes all numbers in either interval, while the intersection includes only numbers that are in both intervals. For example, [1, 5] ∪ [3, 7] = [1, 7] and [1, 5] ∩ [3, 7] = [3, 5].

How do I represent the union of more than two intervals?

You can represent the union of multiple intervals by listing them separated by the union symbol (∪). For example, [1, 3] ∪ [5, 7] ∪ [9, 11] represents the union of three intervals.

What happens when I try to find the union of an empty set with an interval?

The union of an empty set with any interval is simply the interval itself. For example, ∅ ∪ [2, 4] = [2, 4].