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Difference Between an Open Interval & a Closed Interval

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  • 0:04 Open & Closed Intervals
  • 1:21 Types of Intervals
  • 3:30 Interval Notation &…
  • 5:02 Lesson Summary
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Lesson Transcript
Instructor
Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

We'll discuss the difference between an open and closed interval in terms of definition and notation in this lesson. We'll determine if different types of intervals are open and closed and look at how to write them using interval notation.

Open & 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's 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 feet and 6 feet, including 5 feet or 6 feet' or 'strictly taller than 5 feet and strictly shorter than 6 feet (taller than 5 feet, but not including 5 feet, and shorter than 6 feet, but not including 6 feet).


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Both of these alternate signs are examples of intervals in mathematics, where an interval is a range of numbers. Furthermore, the two alternate signs actually represent two different types of intervals.

The sign that reads 'between 5 feet and 6 feet, including 5 feet or 6 feet' is an example of a closed interval. A closed interval is an interval that includes all of its endpoints. 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's easy to recognize that an interval that contains both of its endpoints is closed, and it's easy to recognize that an interval that does not contain both of its endpoints is open. However, sometimes we deal with an interval that contains only one of its endpoints, or an interval that involves infinity. For instance, consider the following intervals:

  1. ax < b or a < xb
  2. -∞ < x < ∞ (all real numbers)
  3. xa or xa
  4. x > a or x < a

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's 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's an open interval.

On the other hand, when an interval involves infinity as an endpoint, it does include all 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.

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Additional Activities

More on Intervals

Consider the closed interval [0, 5] and the open interval (0, 5). They seem pretty similar. But are they really that similar?

Discussion

1) Do closed intervals have a smallest or largest number contained inside the interval? How do you know?

2) Do open intervals have a smallest or largest number contained inside the interval? How do you know?

Using the examples above, [0, 5] and (0, 5) we can see that a closed interval has a smallest and largest number contained inside the interval (in this case, the smallest is 0 and the largest is 5, but in general the smallest and largest numbers in the closed interval are given by the endpoints). We also see that an open interval does not have a smallest or largest number. This is because of the nature of real numbers - try to find a smallest number in the interval (0, 5). It would have to be a number larger than 0, but really close to 0. Try, for example, the number 0.0001. That number is really close to 0, but unfortunately you can always find another real number that is closer, for example 0.000000001 is closer. No matter what number you think of, there is always another real number closer to the end point than the number you thought of. This same concept applies to the largest number in the interval as well.


Why is this true?

Why is it that two intervals that seem so similar can be so different? Think of what the intervals mean. The interval [0, 5] can be thought of as all numbers between 0 and 5 including 0 and 5 and the interval (0, 5) can be thought of as all numbers between 0 and 5 excluding 0 and 5. That is, the closed interval can be represented by 0 ≤ x ≤ 5 and the open interval by 0 < x < 5. That "equal to" is what makes the difference - it ensures a smallest and largest number in the interval, but without the "equal to" there is no smallest or largest number in the interval.


Further Questions

1) Identify the smallest and largest numbers in the interval (7, 93) if they exist.

2) Identify the smallest and largest numbers in the interval [-5, 12] if they exist.

3) Identify the smallest and largest numbers in the interval (3, 17] if they exist.

Solutions

1) This is an open interval - there are no smallest or largest numbers.

2) This is a closed interval, so the smallest number is -5 and the largest number is 12.

3) This is a half-open interval. It has a largest number 17, since the 17 is included in the interval, but it does not have a smallest number, since 3 is not included in the interval.

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