Simplify The Following Union and or Intersection of Intervals Calculator

This calculator helps you simplify the union and intersection of intervals. Whether you're working with numerical ranges, time periods, or other continuous data, this tool will efficiently combine or compare your intervals.

How to Use This Calculator

To simplify intervals using this calculator:

Enter your first interval in the format [a, b] or (a, b)
Enter your second interval in the same format
Select whether you want to calculate the union or intersection
Click "Calculate" to see the simplified result
Review the visualization to understand the relationship between intervals

The calculator will display the simplified interval and show a graphical representation of the operation.

Formula Used

The calculator uses the following formulas for interval operations:

Union of [a, b] and [c, d]: - If b < c or d < a: { [a, b], [c, d] } - Else: [ min(a, c), max(b, d) ] Intersection of [a, b] and [c, d]: - If b < c or d < a: Empty set - Else: [ max(a, c), min(b, d) ]

These formulas account for both closed and open intervals, ensuring accurate results regardless of the interval notation used.

Worked Examples

Example 1: Union of [1, 5] and [3, 7]

Using the union formula: [ min(1, 3), max(5, 7) ] = [1, 7]

Example 2: Intersection of (2, 6) and [4, 8]

Using the intersection formula: [ max(2, 4), min(6, 8) ] = [4, 6]

Example 3: Non-overlapping Intervals

For [1, 3] and [5, 7]: - Union: { [1, 3], [5, 7] } - Intersection: Empty set

Frequently Asked Questions

What is the difference between union and intersection of intervals?

The union of intervals combines all points from both intervals, while the intersection shows only the points common to both intervals.

How do I handle open and closed intervals?

The calculator automatically handles both types of intervals using the standard mathematical definitions for union and intersection.

What if my intervals don't overlap?

For non-overlapping intervals, the intersection will be empty, and the union will show both intervals separately.