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Chapter 0: Problem 50

Give an example of an open interval and a closed interval whose intersection equals the interval (2,5) .

Short Answer

Expert verified
An example of an open interval and a closed interval whose intersection equals the interval (2,5) would be the open interval (1,6) and the closed interval [2,5].

Step by step solution

01

Understand the given interval

We are given the interval (2,5). This is an open interval because it has parentheses (), meaning it does not include its endpoints. Thus, it contains all the real numbers between 2 and 5, excluding 2 and 5.
02

Find an open interval

An open interval whose intersection with another interval equals (2,5), must contain all numbers in the interval (2,5). One way to achieve this is to have an open interval that includes the interval (2,5) entirely. We can choose the interval (1,6) as our open interval. It contains all numbers between 1 and 6, excluding 1 and 6.
03

Find a closed interval

Now we need to find a closed interval that contains the interval (2, 5) and whose endpoints are part of the interval. A closed interval contains its endpoints, represented by square brackets [ ]. We can choose the closed interval [2,5] which contains all real numbers between 2 and 5, including 2 and 5.
04

Check the intersection of the two intervals

Now let's check the intersection of the open interval (1,6) and the closed interval [2,5], taking the common values between the two intervals: Open interval: (1,6) -> Contains values between 1 and 6, excluding 1 and 6. Closed interval: [2,5] -> Contains values between 2 and 5, including 2 and 5. Intersection: (2,5) -> Contains values between 2 and 5, excluding 2 and 5. The intersection of the open interval (1,6) and the closed interval [2,5] is indeed the interval (2,5), as desired. In conclusion, the open interval (1,6) and the closed interval [2,5] have an intersection equal to the interval (2,5).

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