Open and Closed Intervals

Open Intervals and closed intervals are used in mathematics to describe data ranges. Interval symbols and number patterns may be used to help students grasp the distinction between an open interval and a closed interval and the many other forms of intervals. An open or closed interval can represent numeric values.

In case you are looking for open and closed intervals notes, look no further. The article helps you to learn about open and closed intervals, the differences between them, real-world applications, solved cases, and frequently asked questions (FAQs).

What is an Open Interval?

Open Interval is denoted using a pair of two simple brackets ‘()’, also called Parenthesis. An open interval represents the interval alone, which does not contain the endpoints. As a result, we know it won’t cover the whole range of values. As an example, let us consider (a,b). In the example, the values range from a to b. Here, it is impossible to determine the sequence’s beginning or finish, or interval. However, we can assert that the value is somewhere between the stated range, i.e., a and b.

Example of Open Interval

1. Consider (-7,9) as an example of an Open Interval. It can be deduced from the given example that all the values which are bigger than -7 and between 9 are a part of this interval. But the values -7 and 9 are not a part of the Open interval themselves. 

What is a Closed Interval?

In contrast to the definition of the open interval, the set of values that indicate the endpoints are included in closed intervals. The square brackets ‘[ ]’ representing a closed interval indicate that the interval includes the endpoints. Because of this, we can conclude that it will encompass all points in the provided interval. If the set contains the points X=a,b,c,d,e then the set can be expressed in the form of Closed Interval as X= [a,e]. This set has five values, from ‘a’ to ‘e’. The set’s start and final values are included in the closed interval. 

Example of Closed Interval

1. Consider [2,8] as an example for the representation of a Closed Interval. It can be deduced from the example mentioned above that all the values, starting from 2 to 8, i.e., 2, 3, 4, 5, 6, 7, 8, are a part of this interval. The apparent difference is that a closed interval includes both the values mentioned inside the bracket as a part of the range. 

What is the difference between Open Interval and Closed Interval?

The endpoint of the interval helps clarify the distinction between open and closed intervals. The following are some of the differences between the open and closed intervals.

• The endpoints are not included in the open interval but the closed interval.

• () represents the open interval, whereas [] represents the closed interval.

• X < a < y is used to denote an open interval, while x ≤ a ≤ y denotes a closed interval.

• There are two circles at either end of the open interval and a darker dot on the number line at the end of the closed interval.

Application of Open Interval and Closed Interval in Practical Examples

It’s easier to comprehend the difference between open and closed intervals by looking at some simple examples: 

1. Pupils between the ages of 4 and 7 years old are eligible for admission, as are students who are 4 and 7 years old. Students aged 4 and 7 years old and students between the ages of 4 and 7 make up a closed interval.
2. The distinction was awarded to students who scored above 75% on a written test. This is an example of an open interval as the students who have achieved precisely 75%  won’t be awarded a distinction. 
3. The instructor has purchased t-shirts in sizes ranging from 38 to 46. Therefore, the t-shirts in sizes 38 and 46 are included in this closed interval.
4. Informing the patient, the doctor said he would be accessible from Monday to Friday. This is a good illustration of a closed interval since it includes both Monday and Friday in its calculation.
5. An excellent example of an open interval is the phrase 10<x<12, where x only accepts values between 10 and 12, not 10 and 12 in itself.

Conclusion

To put it simply, an interval is concerned with data that cannot be expressed in a precise manner—data that must be expressed in an interval. As part of interval (set-valued) analysis, interval functions are used to deal with uncertainty in intervals, which is a feature of many real-world, deterministic events. As a result, a more specific definition of an interval is a collection of real numbers such that every number y between x and z is likewise a part of the collection. Also, keep following the open and closed intervals notes to stay updated about the latest upgrades in the methods used. And do not forget to go through the open and closed intervals HOTS questions.