Union Ans Intersection of Intervals Calculator

Understanding how to combine and compare intervals is essential in mathematics, computer science, and data analysis. This calculator helps you find the union and intersection of two intervals, providing clear visualizations and step-by-step explanations.

What are Intervals?

An interval is a set of real numbers that lie between two endpoints. Intervals are commonly represented using interval notation, which uses square brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints.

For example:

[a, b] represents all numbers x such that a ≤ x ≤ b
(a, b) represents all numbers x such that a < x < b
[a, b) represents all numbers x such that a ≤ x < b
(a, b] represents all numbers x such that a < x ≤ b

Note: Intervals can also be open (no endpoints), closed (both endpoints included), or half-open (one endpoint included).

Union of Intervals

The union of two intervals is the set of all numbers that belong to either interval. In interval notation, the union is represented by the ∪ symbol.

If you have two intervals [a, b] and [c, d], their union is [min(a, c), max(b, d)] if the intervals overlap or are adjacent. If they don't overlap, the union is the combination of both intervals.

For example, the union of [1, 4] and [3, 7] is [1, 7].

Intersection of Intervals

The intersection of two intervals is the set of all numbers that belong to both intervals. In interval notation, the intersection is represented by the ∩ symbol.

If you have two intervals [a, b] and [c, d], their intersection is [max(a, c), min(b, d)] if the intervals overlap. If they don't overlap, the intersection is the empty set.

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

How to Use the Calculator

Enter the first interval in the format [a, b], (a, b), [a, b), or (a, b].
Enter the second interval using the same format.
Click the "Calculate" button to see the union and intersection results.
Review the visual representation of the intervals.

Tip: The calculator handles both closed and open intervals. Make sure to use the correct brackets for your specific problem.

FAQ

What is the difference between union and intersection?: The union includes all numbers from either interval, while the intersection includes only numbers common to both intervals.
Can I use negative numbers in the intervals?: Yes, the calculator accepts negative numbers in the interval notation.
What if the intervals don't overlap?: If the intervals don't overlap, the intersection will be the empty set, and the union will be the combination of both intervals.
How do I represent an empty interval?: An empty interval is represented as ∅ or an empty set.
Can I use decimal numbers in the intervals?: Yes, the calculator accepts decimal numbers in the interval notation.