An interval consists of that range between two given numbers which includes all the numbers lying between those two numbers. And whether endpoints are included in the interval or not, that will depend on the type of interval.

Types of Intervals

Mainly intervals are of two types but the third type is formed with the help of the first two types.

Following are the three types of intervals:-

1. Open interval
2. Closed interval
3. Half open interval

Open interval: The intervals which do not include endpoints, only numbers between endpoints are included in it, these are known as open intervals.

Open intervals are enclosed by small brackets i.e., (). 

Example, Suppose you have an open interval (3,7)

Then it means that this interval includes all values between 3 and 7, but does not include 3 and 7. Values included are 4,5 and 6 only. 

Closed interval: The intervals which include all values between endpoints and include endpoints also are known as closed intervals.

These types of intervals are enclosed by [].

Example, If an interval is written in this way as [2,6]. Then, which values are included in it?

As above said, these intervals include all values in between and endpoints also. So, this interval includes 2,3,4,5,6.

Half open interval: These intervals are the combination of both open and closed Interval. These intervals are enclosed in (] or [). 

If we have (] this type of interval then the first bracket is open and the second one is closed, it means that we shall not include the first endpoint but we have to include the value between endpoints and last endpoint value.

Example, you have an interval (3,5], then which values does it consist of?

This interval includes 4 and 5.

As, 4 is between endpoints and 5 is that endpoint which is at the closed interval side bracket.

And, if you have [) this type of bracket, then it means that you have to include first endpoint and values between endpoints but not include second endpoint.

For example, you have an interval [1,5), then what does this mean?

This interval gives us the value only 1,2,3,4 and will not give 5 as its value.

Points to remember

Endpoint number which is having a small bracket, that value will not be included and value at the side of the big bracket will surely include.

Including or excluding endpoints

• To suggest that one of the endpoints is to be excluded from the set, the corresponding rectangular bracket can be either replaced with a parenthesis, or reversed. Both notations are defined in  International standard ISO 31-11. As a result, in set builder notation, Each interval (a, a), [a, a), and (a, a] defined  the empty set, whereas [a, a] represents the singleton set {a}. When a > b, all four notations are usually taken to represent the empty set.
• Each notation may additionally overlap with other uses of parentheses and brackets in mathematics. For instance, the notation (a, b) is regularly used to denote an ordered pair in set theory.

How to Define an Interval

For any two numbers a and b, where a  ≤ b, the set {x: a ≤ x ≤ b} is defined as an interval.

And when a<b, the set {x: a < x < b} is defined as an interval. The difference between these two intervals is only that the first interval is including both endpoints and the second interval is not including endpoints. 

Some intervals do not have a finite endpoint, these sets are of the following types:-

(a, ∞) is the interval {x: a < x}.

(−∞, a] is the interval {x: x ≤ a}.

(−∞, ∞) is the set of all numbers

Importance of Intervals

Here are few important points of intervals are written below-

• Real intervals play a vital role within the concept of integration because they are the most effective units whose “size” (or “measure” or “duration”) is straightforward to outline. The idea of measure can then be extended to greater complex sets of actual numbers, leading to the Borel measure and subsequently to the Lebesgue degree.
• Intervals are principal to interval arithmetic, a preferred numerical computing method that automatically gives guaranteed enclosures for arbitrary formulas, even within the presence of uncertainties, mathematical approximations, and mathematics round off.

Point to remember

When you are having infinity in the interval, then always remember that you have to put an open bracket at the infinity side always. If you put a close bracket then your solution will be wrong.

Conclusion

In this article, you have read about intervals, their types, and definitions along with a few examples.

Intervals are considered as the subset of real numbers, which consists of the numbers between any two real numbers. If you are having a close interval, then you have to include endpoints and if you are having an open interval then you have to take only those values which are present between those two endpoints. We have discussed the significance of the interval.