Open Interval and Closed Interval

Open interval and closed interval are used in mathematics to represent a range of real numbers present in between two end points. They define whether the boundary points or the endpoints are included in the interval or not. 

An open interval includes all the numbers between two end points but does not include the end points themselves. It is written in the form (a, b), which means all values x such that a < x < b. On a number line, open intervals are shown using hollow circles at the end points, to indicate that those points are not part of the interval. 

A closed interval, on the other hand, includes every number between two endpoints, including the endpoints themselves. It is written with square brackets, in the form [a, b], which means all values x such that a ≤ x ≤ b. On a number line, closed intervals use solid circles to show that the boundary values are included in the interval. 

For example, (-3, 5) represents all real numbers between -3 and 5, excluding -3 and 5, while [2, 7] includes the total range from 2 to 7, including the endpoints. 
 

In open intervals, numbers between the endpoints are enclosed within 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 are 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.

Open Interval Notation
 

An open interval from a to b is written as (a, b). This means all real numbers x such that a, x < b. In set-builder notation, this is written as:
{x ∈ R ∣ a < x < b}.

Open Interval on a Number Line
 

On a number line, an open interval is shown using hollow (unfilled) circles at the endpoints. These hollow circles indicate that the endpoints are not included.
The space between the hollow circles represents all the numbers between a and b. The endpoints are excluded. 

The hollow circles at a and b show that the endpoints are excluded in the interval.

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.

Closed Interval Notation 
 

Closed intervals are written using square brackets [ ], in the form [a, b]. This means that a is included, b is included, and every number between them are also included. Closed intervals are represented as { x ∈ R ∣ a ≤ x ≤ b }.

Closed Interval on a Number Line? 

On a number line, closed intervals are shown using solid circles at the endpoints. These solid dots indicate that the endpoints are included in the interval. 

This means, all values from a to b, including a and b are part of the interval.

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. These operations are similar to those 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 = \({({a_1}, {b_1}) }\)and B = \({({a_2}, {b_2})}\)

The union of A and B is:

\({{A \cup B} = \{\, x \in \mathbb{R} \mid a_1 < x < b_1 \ \text{or} \ a_2 < x < b_2 \,\} }\)

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 \(A = {({a_1}, {b_1})}\) and \(B = {({a_2}, {b_2})}\)

The intersection of A and B is
\(A \cap B = {\{\, x \in \mathbb{R} \mid \max(a_1, a_2) < x < \min(b_1, b_2) \,\} }\)

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 \(A^c = (-\infty, a] \cup [b, \infty) \)

For example, \(A = (2, 5)\)

\(A^c = (-\infty, 2] \cup [5, \infty) \)

→ \((-∞, 2]\) includes all numbers that are less than or equal to 2

→ \([5, ∞)\) includes all numbers that are greater than or equal to 5

Learning open and closed intervals becomes easy when students understand their symbols, number line representations, and real-life examples. In this section, we will learn some tips and tricks to master open intervals and closed intervals.

• Remember the brackets, that is, the open intervals use parentheses () and the closed interval is represented using the square bracket [].  
   
• On a number line, a hollow circle shows an open interval, signaling that the endpoint is not part of the set. A filled circle represents a closed interval, meaning the endpoint is included.
   
• Use a number line to visualize the interval to see the endpoints clearly. This visual approach helps you instantly recognize whether it’s open or closed.
   
• Open intervals correspond to strict inequalities \({a < x < b}\), while closed intervals correspond to inclusive inequalities \({a ≤ x ≤ b}\). This makes it quick to write or recognize intervals.
   
• Start with simple intervals like (2, 5) and [2, 5], then move on to decimals, fractions, and negative numbers to strengthen your understanding. 
   
• Encourage children to explain open and closed intervals in their own words, this will help them in strengthening the understanding of the concepts.
   

• Use real-life examples, such as temperature ranges, age limits or sports timing, to show where open and closed intervals appear naturally.  This will help students to relate math to everyday situations. 
   

• Parents and teachers can provide hands-on practice using number line strips, stickers or so on. By this, students can physically mark open and closed intervals and the endpoints with right understanding.
   
• Introduce to students simple error-checking methods like asking them “are the endpoints included here?” This helps them in self-correction and concept clarity.
   
• Use digital tools or interactive apps to help students experiment with intervals with engagement. These are effective for visual learning too. 

Intervals are used in daily life to represent range of values such as time, measurements, and prices. It is important to know how intervals are used in everyday life. Given below are some real-life applications.
 

• Scheduling time: Intervals help us indicate when an event starts and ends. For example, an event scheduled from \([2:00\ \text{pm}, 2:00\ \text{pm}] \)means it includes both 2:00 pm and 5:00 pm. If the interval were open, \((2:00\ \text{pm}, 5:00\ \text{pm}) \), the event would start just after 2:00 pm and end just before 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: Companies often specify salary ranges for roles using intervals, for example, \([\$30{,}000; \$50{,}000] \) per month. This indicates that salaries can include the minimum and maximum values. An open interval could be used to indicate salaries strictly above or below a certain value.
   
• Temperature limits: Weather reports or manufacturing processes often specify safe or optimal temperature ranges. For instance, a chemical may be stable in the range \([20^\circ \text{C}, 50^\circ \text{C}] \), meaning temperatures at \({20^\circ \text{C}}\)and \({50^\circ \text {C}}\) are safe. If temperatures \((20^\circ \text{C}, 50^\circ \text{C}) \) are considered, the exact endpoints are avoided because they may cause instability.
   
• Age restrictions: Age limits for certain activities or legal requirements often use intervals. For example, a roller coaster may allow riders aged [5, 12] years, including 5 and 12. A vaccine dosage recommendation might use an open interval like (6 months, 12 months), meaning exactly 6 or 12 months may not meet the criteria.