Cal11 calculator
Search calculator...

Union and Intersection of Intervals Calculator Interval Notation

Reviewed by Calculator Editorial Team

This calculator helps you find the union and intersection of intervals using interval notation. Interval notation is a concise way to represent sets of real numbers, making it easier to visualize and compute combined or overlapping intervals in mathematics.

What is Interval Notation?

Interval notation is a mathematical way to represent a set of real numbers that lie between two endpoints. It's commonly used in calculus, algebra, and analysis to describe ranges of values.

There are four main types of intervals:

  • Closed interval: [a, b] - includes all numbers from a to b, including a and b
  • Open interval: (a, b) - includes all numbers from a to b, excluding a and b
  • Half-open (half-closed) interval: [a, b) or (a, b] - includes one endpoint but not the other
  • Infinite interval: [a, ∞) or (-∞, b] - includes all numbers from a to infinity or negative infinity to b

Note: The square brackets [ ] indicate that the endpoint is included, while parentheses ( ) indicate that the endpoint is excluded.

Union of Intervals

The union of two intervals is the set of all elements that are in either interval. In interval notation, the union is represented by the symbol ∪.

A ∪ B = {x | x ∈ A or x ∈ B}

For example, the union of [1, 3] and [2, 4] is [1, 4].

Rules for Union of Intervals

  1. If the intervals overlap or touch, combine them into a single interval.
  2. If there's a gap between intervals, keep them separate in the union.
  3. For infinite intervals, the union will be the smallest interval that covers both.

Intersection of Intervals

The intersection of two intervals is the set of all elements that are in both intervals. In interval notation, the intersection is represented by the symbol ∩.

A ∩ B = {x | x ∈ A and x ∈ B}

For example, the intersection of [1, 5] and [3, 7] is [3, 5].

Rules for Intersection of Intervals

  1. If the intervals overlap, the intersection is the overlapping part.
  2. If there's no overlap, the intersection is the empty set (∅).
  3. For infinite intervals, the intersection will be the overlapping finite part.

How to Use This Calculator

  1. Enter the first interval in the format [a, b], (a, b), [a, b), or (a, b]. Use ∞ for infinity.
  2. Enter the second interval using the same format.
  3. Click "Calculate" to see the union and intersection results.
  4. View the results and chart visualization.
  5. Use the "Reset" button to clear the form.

Tip: For infinite intervals, use ∞ or -∞ as one of the endpoints. For example, [5, ∞) or (-∞, 10].

FAQ

What is the difference between union and intersection of intervals?

The union of intervals includes all numbers that are in either interval, while the intersection includes only numbers that are in both intervals. Union is represented by ∪ and intersection by ∩.

How do I represent an empty set in interval notation?

An empty set is represented by ∅ or sometimes as an interval that doesn't contain any numbers, like [3, 2] which is impossible since 3 > 2.

Can I use this calculator for more than two intervals?

This calculator is designed for two intervals at a time. For more complex operations with multiple intervals, you would need to perform operations sequentially.

What happens if I enter invalid interval notation?

The calculator will display an error message if the interval notation is invalid. Make sure to use proper brackets and numbers in the correct format.