Last updated on July 23rd, 2025
An interval is a mathematical concept. Often written in pairs, intervals can be used to enclose a series of numbers between two endpoints, represents the values, excluding the endpoints. In a closed interval, the end points are also included. In this article, we will learn more about them.
In open intervals, numbers between the endpoints are written within the parentheses. For example, the open interval of (2, 5) includes all real numbers between 2 and 5 but not the endpoints (2 and 5). Here, 2 and 5 are the endpoints and not included in the open interval. The general way of representing an open interval is a < x < b, where a and b are the endpoints. In set notation, open intervals are represented as {x ∈ R | a < x < b}. Let’s consider an open interval (2,5). Therefore, the set notation will be {x ∈ R | 2 < x < 5}.
An open interval on a number line is shown by making use of the hollow circles at the endpoints. This means that the endpoints are excluded.
In closed intervals, we include the endpoints and the numbers between them. They are represented using the [] brackets. For example, in [-4, 4] we represent all real numbers from -4 to 4. The general way of representing a closed interval is a ≤ x ≤ b, where a and b are the endpoints to be included. In set notation, closed intervals are represented as {x ∈ R | a ≤ x ≤ b}. Let’s consider [-4, 4] as an example. Therefore, the set notation will be {x ∈ R |-4 ≤ x ≤ 4}.
A closed interval on a number line is shown by making use of the filled circles on the endpoints. This means that the endpoints are included.
Now that we have learned about open and closed intervals, let us try to understand the difference between them. Given below is a table showing their differences:
Open Interval | Closed Interval |
Represented using () brackets | Represented using [] brackets |
Do not include the endpoints | Endpoints are included |
On the number line, an open interval is shown with the help of hollow circles | A closed interval is represented by using filled circles on the number line |
Generally represented as a < x < b | Generally represented as a ≤ x ≤ b |
Various operations can be performed on intervals, such as union, intersection, and complement. They are the same as the ones performed on sets. Let’s look at them in detail.
The union of intervals ‘A’ and ‘B’ includes all elements of A and B.
If A = (a1, b1) and B = (a2,b2)
The union of A and B is:
A∪B ={x∈R∣a1<x<b1 or a2<x<b2}
For example, let A be (1, 5) and B be (3, 9)
A∪B = (1, 9)
Since intervals (1, 5) and (3, 9) overlap, their union is the open interval (1, 9)
If A = [2, 8] and B = (9, 12), the intervals will be disjoint because 8 is less than 9.
The union A∪B = [2, 8] ∪ (9, 12) will include numbers from 2 to 8 and from 9 to 12. This union includes 2 and 8 and excludes 9 and 12.
The intersection of intervals ‘A’ and ‘B’ contains common elements of A and B.
If A = (a1, b1) and B = (a2, b2)
The intersection of A and B is
A∩B = {x ∈ R∣ max(a1,a2) < x < min(b1, b2)}
Check the examples given below:
A = (1, 4) and B = (2, 7)
‘A’ includes numbers between 1 and 4, and ‘B’ includes numbers between 2 and 7
Therefore, A∩B = (2, 4)
A = [5, 10] and B = (6,15)
A is a set of numbers from 5 to 10. Whereas, B is a set that includes numbers between 6 and 15.
Therefore, A∩B = [6, 10]
The complement of an interval includes all real numbers not in the interval.
If A = (a, b),
The complement of A will be Ac = (-∞, a] ∪ [b, ∞)
For example, A = (2, 5)
Ac = (-∞, 2] ∪ [5, ∞)
→ (-∞, 2] includes all numbers that are less than or equal to 2
→ [5, ∞) includes all numbers that are greater than or equal to 5
In our everyday life, intervals are used to represent time, measurements, or prices. It is important to know how intervals are used in everyday life. Given below are some real-life applications.
Students get confused with open and closed intervals. Such misunderstandings can lead to incorrect results. By identifying common mistakes, students can better understand intervals:
Find the union of intervals if A = [1, 4] and B = (3, 7)
A∪B = [1, 7)
Union contains all numbers from both the intervals. Since A is closed, 1 is included; and since B is open, 7 is excluded. Therefore, the union of intervals A and B will be:
A∪B = [1, 7)
What will be the complement of interval A, if A = (5, 20)?
(-∞, 5] ∪ [20, ∞)
The complement of an interval consists of real numbers except those in the interval. Therefore, the complement of A will be:
Ac = (-∞, 5] ∪ [20, ∞)
This is represented by all numbers less than or equal to 5, or greater than or equal to 20.
What will be the intersection of the intervals A = [2, 6] and B = (4, 8)
A ∩ B = (4, 6]
The common numbers between [2, 6] and (4, 8) are 4 to 6.
Therefore, the intersection of A and B will be:
A ∩ B = (4, 6]
Find the intersection of A = (1, 5) and B = (5, 10)
A ∩ B = ∅
Since neither of the intervals includes 5, A ∩ B will be empty.
Hence, A ∩ B = ∅
What will be the union of (-∞, 0] and [0, 3]?
(-∞, 3]
Since 0 is also included in the second interval, they can be linked together. Hence, the union of (-∞, 0] and [0, 3] is (-∞, 3]
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.