We will discuss the difference between an open and closed interval in terms of definition and notation.
We will determine if different types of intervals are open and closed and look at how to write them using interval notation.
Open and Closed Intervals
Imagine this: Sheila and her friend, Harry, are at an amusement park in line for a ride where they see a sign that reads ‘You must be between 5 and 6 feet tall to ride this ride.’ Well, this presents a question! You see, Sheila is exactly 5 feet tall, and Harry is exactly 6 feet tall, so can they ride the ride or not?
|This is an example of why it is important for intervals of numbers to specify whether or not they include their endpoints. In this case, it would have been much more helpful if the sign read either ‘between 5 ft and 6 ft, including 5 ft or 6 ft’ or ‘strictly taller than 5 ft. and strictly shorter than 6 ft.
(taller than 5 ft, but not including 5 ft. and shorter than 6 feet, but not including 6 feet).
On the other hand, the sign that reads ‘between 5 and 6 feet, but not including 5 feet and 6 feet’ is an example of an open interval, where an open interval is an interval that does not contain its endpoints.
Types of Intervals
It is easy to recognize that an interval that contains both of its endpoints is closed, and it is easy to recognize that an interval that does not contain both of its endpoints is open. However, sometimes we deal with intervals that contains only one of its endpoints or an interval that involves infinity. For instance, consider the following intervals:
In the first intervals, we see that the intervals include one endpoint, but not the other. When this is the case, we don’t classify the interval as open or closed, we say that it is a half-open interval or a half-closed interval. We use these two terms interchangeably.
The second interval involves infinity. We can look at infinity and negative infinity as endpoints in two ways. On one hand, infinity is a concept, not an actual number, so we can’t ever actually reach it. Viewing it this way, we would say the endpoints infinity and negative infinity are not included in the interval, so it is an open interval.On the other hand, when an interval involves infinity as an endpoint, it does include all of the numbers up to it, and since infinity and negative infinity go on forever, the interval does include all of its endpoints. Viewing it this way, we would say the interval is closed.
Confused yet? Basically, we say that the interval, -; ; x ; ; is both open and closed, and an endpoint of infinity or negative infinity can be thought of as included or not included in an interval.Because of this, when we have an interval that has a number as one of its endpoints and infinity or negative infinity as its other endpoint, as in the third and fourth intervals listed, we can say that if the number endpoint is included in the interval, then the interval is closed (as in the third examples), and if the number endpoint is not included in the interval, then the interval is open (as in the fourth examples).
Interval Notation, Number Lines, and Examples
Often, intervals of numbers will be expressed using interval notation. To represent an interval using interval notation, we write the endpoints, and separate them by a comma.
If the endpoint is included, we use a bracket to indicate this, and if a endpoint is not included, we use parentheses to indicate this. When infinity is an endpoint, we always use parentheses.
Since it includes its endpoints, it is a closed interval. For the interval 3 ; x ; 10, we would write (3, 10). Since it does not include its endpoints, it is an open interval.Consider the interval x ; -2. First of all, is this interval open, closed, or both? If you are thinking that it is open, then congrats! You are right! It has infinity as one endpoint, and it doesn’t include its other endpoint, -2, so it is an open interval. Now, let’s represent it using interval notation. We first write -2 and ; separated by a comma:
Next, we simply determine whether to use a bracket or a parentheses at each of these endpoints.
We are not including the endpoint -2, so we use a parentheses at this endpoint, and since we always use parentheses at infinity or negative infinity, we also use a parentheses at ;. All together, we have the following:
Not so hard, is it?
An interval is a range of numbers. A closed interval, a, b, is an interval that includes all of its endpoints, and an open interval, (a, b), is an interval that does not contain its endpoints.
When an interval involves infinity or negative infinity, we have the following rules for whether it is an open or closed interval.
As we saw in the example of the amusement park ride, it is important to know whether intervals of numbers are open or closed in order to correctly analyze the situation the interval is being used in.
Therefore, we should definitely retain this knowledge for future use.