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 is Open Interval and Closed Interval?
An open interval does not involve its end values, whereas a closed interval involves it 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 circle at the end points, whereas the closed intervals are represented by filled or darkened circle at the end points.
Open Interval Definition
An open interval is an interval which does not contain the end points. It is represented by parenthesis (). An example of open interval is (a, b), where a and b are not included.
Open Interval Example
An example of open interval is a number greater than -2 and less than 3 i.e., 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 shows the open interval representation in 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.
|
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. |
Also, Check
FAQs on Open Interval and Closed Interval
What is the Difference Between an Open Interval and a Closed Interval?
The Difference between an open interval and a closed interval is that the end points are not included in the open interval whereas the endpoints are included in the closed Interval.
What is the Definition of Closed Interval?
The definition of closed interval is the interval that includes the limits i.e., endpoints.
What is an Example of Open Interval?
An example of an open interval is that if there is an interval containing numbers greater than 3 and less than 7 then, it is an open interval represented as (3, 7).
What is the Symbol of Open Interval?
The symbol of open interval is parenthesis. It is written as (a, b).
What is the Symbol of Closed Interval?
The symbol of closed interval is [a, b]
What is Semi-Open or Semi-Closed Interval?
Semi-Open or Semi-Closed interval are those in which one end point is included and other is excluded
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