Problem 49 Give an example of an open inter... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 0: Problem 49

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

Short Answer

Expert verified

An example of an open interval and a closed interval whose union equals the interval (2,5) is the open interval (2,4) and the closed interval [4,5). The union of these two intervals, (2,4) 鈭� [4,5), contains all the points x such that 2 < x < 5, which is equivalent to the interval (2,5).

Step by step solution

Understanding the interval notation

First, let's understand the interval notation. The interval (a, b) represents all points x such that a < x < b. It is called an open interval. The interval [a, b] represents all points x such that a 鈮� x 鈮� b. It is called a closed interval.

Selecting intervals

Let's select an open interval and a closed interval such that their union equals the interval (2,5). We can choose an open interval that lies within the given interval, for example, (2,4), and a closed interval that covers the rest of the given interval, for example, [4,5].

Verifying the union

Now, let's check whether the union of the open interval (2,4) and closed interval [4,5] equals the given interval (2,5). The union of two intervals is all the points that belong to either of them. In this case, the union of (2,4) and [4,5], denoted by (2,4) 鈭� [4,5], contains all the points x such that 2 < x < 4 or 4 鈮� x 鈮� 5. Notice that 2 < x 鈮� 5, which is equivalent to the interval (2,5]. However, since the question asks us to find a union that equals the interval (2,5) and not (2,5], we need to modify the chosen intervals.

Modifying the chosen intervals

Recall that the closed interval includes the boundaries, while the open interval does not. Therefore, we could create a new closed interval that does not include the right endpoint 5. Let's modify the chosen intervals and define a a new closed interval as [4,5), and keep the same open interval as (2,4).

Verifying the new union

Check whether the union of the new open interval (2,4) and the new closed interval [4,5) equals the given interval (2,5). The union of (2,4) and [4,5), denoted by (2,4) 鈭� [4,5), contains all the points x such that 2 < x < 4 or 4 鈮� x < 5. Notice that 2 < x < 5, which is equivalent to the interval (2,5). So, the open interval (2,4) and closed interval [4,5) satisfy the requirement, and their union equals the interval (2,5).

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91影视

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

Short Answer

Step by step solution

Understanding the interval notation

Selecting intervals

Verifying the union

Modifying the chosen intervals

Verifying the new union

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