Interval Notation Calculator Union

Interval notation is a way to represent sets of real numbers using parentheses and brackets. 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 way to represent a range of real numbers. It uses brackets and parentheses to indicate whether the endpoints are included or excluded from the interval.

Key symbols in interval notation:

[a, b] - Includes all numbers from a to b, including a and b
(a, b) - Includes all numbers from a to b, excluding a and b
[a, b) - Includes all numbers from a to b, including a but excluding b
(a, b] - Includes all numbers from a to b, excluding a but including b

Interval notation is commonly used in calculus, algebra, and other branches of mathematics to describe domains, ranges, and solution sets.

How to Find the Union of Intervals

The union of two intervals combines all numbers that belong to either interval. To find the union of two intervals [a, b] and [c, d], follow these steps:

Compare the lower bounds (a and c) to find the smallest value
Compare the upper bounds (b and d) to find the largest value
Determine whether the endpoints should be included or excluded based on the original interval notation
Write the resulting interval in proper notation

Union of two intervals [a, b] and [c, d]:

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

Otherwise, the union is the set of two separate intervals

If the intervals do not overlap and are not adjacent, the union is simply the set of both intervals written separately.

Examples of Interval Union

Let's look at some examples to understand how interval unions work.

Example 1: Overlapping Intervals

Find the union of [1, 5] and [3, 7].

The intervals overlap from 3 to 5. The union combines the entire range from 1 to 7.

Result: [1, 7]

Example 2: Adjacent Intervals

Find the union of [2, 4] and [4, 6].

The intervals are adjacent at 4. The union combines the entire range from 2 to 6.

Result: [2, 6]

Example 3: Separate Intervals

Find the union of [1, 3] and [5, 7].

The intervals do not overlap or touch. The union is the set of both intervals.

Result: [1, 3] ∪ [5, 7]

Note: The union symbol (∪) is used when the intervals are separate and do not overlap.

FAQ

What is the difference between union and intersection in interval notation?

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, 5] and [3, 7] is [1, 7], while the intersection is [3, 5].

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

For more than two intervals, you can continue to use the union symbol (∪) between each interval. For example, the union of [1, 3], [5, 7], and [9, 11] would be written as [1, 3] ∪ [5, 7] ∪ [9, 11].

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

The union of any interval with the empty set is simply the original interval. For example, [1, 5] ∪ ∅ = [1, 5].