Interval Notation Union of Two Sets Calculator

What is Interval Notation?

Interval notation is a way to represent a set of real numbers that fall between two endpoints. It's commonly used in calculus, analysis, and other mathematical fields. The most common types of intervals are:

• 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])
• Infinite interval: Extends to infinity (e.g., [a, ∞) or (-∞, b])

Interval notation provides a concise way to describe ranges of numbers without listing each individual element.

How to Find the Union of Two Sets

The union of two sets A and B, denoted as A ∪ B, is the set of all elements that are in A, in B, or in both. When working with intervals, the union combines the ranges of both intervals.

Steps to Calculate the Union

1. Identify the lower and upper bounds of both intervals
2. Determine the smallest lower bound and largest upper bound
3. Combine the intervals, adjusting the brackets based on whether the endpoints are included or excluded
4. Simplify the result if possible (e.g., combining overlapping or adjacent intervals)

For interval notation, the union of [a, b] and [c, d] is [min(a, c), max(b, d)] if the intervals overlap or are adjacent. If they don't overlap, the result is the combination of both intervals.

Examples of Union Calculations

Example 1: Overlapping Intervals

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

The intervals overlap from 3 to 5. The union is [1, 7].

Example 2: Adjacent Intervals

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

The intervals are adjacent at 4. The union is [2, 6].

Example 3: Non-overlapping Intervals

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

The intervals don't overlap. The union is [1, 3] ∪ [5, 7].

Example 4: Different Interval Types

Find the union of [1, 4) and (3, 7].

The first interval includes 1 and 4, while the second excludes 3 and 7. The union is [1, 7].