Open Interval and Closed Interval

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. In set notation, open intervals are represented as \(\{\, x \in \mathbb{R} \mid a < x < b \,\} \). 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 \(\{\, x \in \mathbb{R} \mid a \le x \le b \,\} \). Let’s consider [-4, 4] as an example.  Therefore, the set notation will be \(\{\, x \in \mathbb{R} \mid -4 \le x \le 4 \,\} \).

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. 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. 

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.