Interval notation is the standard language for describing sets of real numbers in mathematics. Beyond just writing down these sets, we often need to perform operations on them, such as finding which numbers they have in common or combining them into a larger set. Our Interval Notation Calculator is designed to perform these two fundamental set operations: finding the intersection and the union of two intervals. This tool is perfect for students of algebra and set theory to visualize how intervals interact and to quickly verify the results of set operations.
How to Use the Interval Notation Calculator
Performing set operations on your intervals is simple:
- Enter Interval 1: Input your first interval using standard notation (e.g.,
[1, 5)or(-2, 3]). Use parentheses for open endpoints and brackets for closed endpoints. - Enter Interval 2: Input your second interval using the same notation.
- Calculate: Click the "Calculate" button.
- View the Results: The calculator will display the resulting intervals for both the intersection (∩) and the union (∪) of the two sets you entered.
Understanding Set Operations on Intervals
The two most basic operations you can perform on sets are intersection and union. Visualizing the intervals on a number line is often the easiest way to understand the result.
Intersection (∩): The "And" Operation
The intersection of two sets, denoted by the symbol ∩, is the set of all elements that are in both Set A and Set B. When working with intervals on a number line, the intersection is simply the region where the two intervals overlap.
Example: Let's find the intersection of A = [1, 5) and B = (3, 7].
- Interval A includes all numbers from 1 up to (but not including) 5.
- Interval B includes all numbers from 3 (not including 3) up to and including 7.
- The region where they overlap starts just after 3 and ends just before 5.
Therefore, the intersection A ∩ B is (3, 5).
If the two intervals do not overlap at all, their intersection is the "empty set," denoted by the symbol ∅.
Union (∪): The "Or" Operation
The union of two sets, denoted by the symbol ∪, is the set of all elements that are in either Set A or Set B (or both). It is the total combination of all numbers contained within the two intervals.
Using the same example: Let's find the union of A = [1, 5) and B = (3, 7].
- We want to combine all the numbers from both sets into one larger set.
- The combined set starts at the lowest endpoint of either interval, which is 1 (and it's included).
- The combined set ends at the highest endpoint of either interval, which is 7 (and it's included).
Because the intervals overlap, they form a single, continuous new interval. Therefore, the union A ∪ B is [1, 7].
If the two intervals do not overlap, their union is simply written with the union symbol connecting the two original intervals, for example: [1, 2] U [5, 6].
Frequently Asked Questions
What do the parentheses and brackets mean again?
Parentheses ( ) mean the endpoint is *not* included in the interval (an open endpoint). Brackets [ ] mean the endpoint *is* included in the interval (a closed endpoint). This corresponds to the inequality symbols < > versus ≤ ≥.
What if one interval is completely inside another?
Let's say Interval A = [2, 4] and Interval B = [1, 10].
Intersection: Since all numbers in A are also in B, their overlap (intersection) is simply Interval A itself: [2, 4].
Union: Since all numbers in A are already contained within B, their total combination (union) is simply the larger interval, B: [1, 10].
How does this relate to solving compound inequalities?
This is directly related. Solving a compound inequality with "and" (like -2 < x ≤ 5) is the same as finding the intersection of the two inequalities x > -2 and x ≤ 5. Solving a compound inequality with "or" (like x < 0 or x ≥ 3) is the same as finding the union of the two solution sets.