Open Interval and Closed Interval
Last Updated :
23 Jul, 2025
Open Interval and Closed Interval: A closed interval includes its endpoints and is enclosed under square brackets. An open interval does not include its endpoints and is enclosed under parenthesis. In this article, we will explore the difference between an open interval and a closed interval, along with the open interval and closed interval definitions.
Let's start our learning on the topic "Difference Between an Open Interval and a Closed Interval".
What are Open Interval and Closed Interval?
An open interval does not involve its end values, whereas a closed interval involves its end values. The open interval is represented by the parenthesis and the closed intervals are represented by square brackets. In the number line representation, the open interval is represented by a circle at the endpoints, whereas the closed intervals are represented by a filled or darkened circle at the endpoints.
Open Interval Definition
An open interval is an interval that does not contain the endpoints. It is represented by parenthesis (). An example of the open interval is (a, b), where a and b are not included.
Open Interval Example
An example of an open interval is a number greater than -2 and less than 3 i.e., an open interval (-2, 3). The open interval (-2, 3) includes the numbers from -2 to 3 where -2 and 3 are excluded.
Open Interval in Number Line Representation
The open interval in the number line is represented by a circle. Below number line below shows the open interval representation in the number line.
Closed Interval Definition
A closed interval is the interval which contains the end points. It is represented by square brackets. An example of closed interval is [a, b] where a and b are included.
Closed Interval Example
An example of closed interval is numbers greater than or equal to -2 and less than or equal to 3 i.e., closed interval [-2, 3]. The closed interval [-2, 3] includes the numbers from -2 to 3 where -2 and 3 are included.
Closed Interval in Number Line Representation
The closed interval in the number line is represented by a filled circle. Below number line shows the closed interval representation in number line.
Operations on Open and Closed Interval
The operations on open and closed intervals are:
- Union
- Intersection
- Complement
Union of Intervals
The union of two intervals P and Q contains all the elements present in one or both the intervals.
Examples
Some examples of union of two intervals are given below.
- (2, 7) ∪ (4, 9) = (2, 9)
- [1, 10] ∪ [5, 15] = [1, 15]
- (5, 12) ∪ [6, 14] = (5, 14]
Intersection of Intervals
The intersection of two intervals P and Q contains the elements present in both the intervals.
Examples
Some examples of union of two intervals are given below.
- (2, 7) ∩ (4, 9) = (4, 7)
- [1, 10] ∩ [5, 15] = [5, 10]
- (5, 12) ∩ [8, 14] = [8, 12)
Complement of Interval
The complement of interval contains the elements that are not included in the interval.
Examples
Some examples of complement of interval are given below.
- X = [3, 6] then, X' = (-∞, 3) ∪ (6, ∞)
- Y = (4, 11) then, Y' = (-∞, 4] ∪ [11, ∞)
Difference Between Open Interval and Closed Interval
The below table represents the difference between an open interval and a closed interval.
Difference Between Open Interval and Closed Interval |
|---|
Characteristics | Open Interval | Closed Interval |
|---|
Definition | An interval that does not include the endpoints is called an open interval. | An interval that includes the endpoints is called a closed interval. |
|---|
Brackets | Open interval uses parenthesis. | Closed interval uses square brackets. |
|---|
Representation | It is represented as (a, b). | It is represented as [a, b]. |
|---|
Example | An example of open interval is 2 < x < 9. | An example of closed interval is 2 ≤ x ≤ 9. |
|---|
Number Line Representation | An open interval is represented as a circle at the endpoints in a number Line. | A closed interval is represented as a filled circle at endpoints in a number Line. |
|---|
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Open Interval and Closed Interval Solved Examples
Example 1:
Find the intersection of the intervals (2,5) and [4,7].
Solution:
The interval (2,5) includes numbers between the 2 and 5 excluding 2 and 5.
The interval [4,7] includes numbers between the 4 and 7 including 4 and 7.
The intersection is [4,5] which includes numbers between the 4 and 5 including 4 but not 5.
Example 2:
Determine if the number 3 is in the interval [2,6].
Solution:
The interval [2,6] includes numbers between 2 and 6 including the 2 but excluding 6.
Since 3 is between 2 and 6, it is included in the interval [2,6].
Example 3:
Find the union of the intervals [1,4] and [3,5].
Solution:
The interval [1,4] includes numbers between the 1 and 4 excluding the 1 but including 4.
The interval [3,5] includes numbers between the 3 and 5 including the 3 but excluding 5.
The union is (1,5) which includes numbers between the 1 and 5 excluding 1 and 5.
Example 4:
Determine the length of the interval [7,12].
Solution:
The interval [7,12] includes numbers between the 7 and 12 including both the endpoints.
The length of the interval is 12−7=5.
Example 5:
Find the complement of the interval (3,8) in the real numbers.
Solution:
The complement includes all real numbers not in the interval (3,8).
The complement is (−∞,3 ]∪[ 8, ∞).
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Practice Problems on Open Interval and Closed Interval
1. Find the intersection of the following intervals: A = (1,5) and B = [3,7]
2. Determine whether the following statement is true or false: (0,4) ∪ (4,8) = (0,8)
3. Find the union of the following intervals: A = [2,6) and B = (4,8]
4. Given the interval C=[a,b) express the interval C in terms of its boundary points a and b. Explain the meaning of the notation in the context of open and closed intervals.
5. Determine the length of the interval D = [2,9). How does the length change if the interval were instead D′ = (2,9]
Conclusion - Open Interval and Closed Interval
The Open and closed intervals are fundamental concepts in the mathematics particularly in the calculus and real analysis. An open interval excludes its endpoints while a closed interval includes them. Understanding these intervals is crucial for the solving problems involving ranges of the values domain and range of functions and integration. The Mastery of the interval notation enhances problem-solving skills and clarity in the mathematical communication.
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