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.