Open Interval and Closed Interval

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:

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.

• Scheduling time: Tells us when an event is scheduled to start and end. The interval can be represented as [2:00 pm, 5:00 pm]. This means the interval includes 2:00 pm and 5:00 pm.

• Education: To find who failed and who passed the exam. Scores with the interval [50%, 100%] are considered passing, but anything below 50% (open interval) is considered failing. This means a score of 50% is passing and a score of 49% is failing.

• Salary: The intervals are used to show the salary range for different job roles offered by the companies.