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Chapter 3: Problem 1

\(S=\\{x: 0

Short Answer

Expert verified
The set S corresponds to the interval (0, 3].

Step by step solution

01

Understanding the Set Notation

The notation used here is a set builder notation, which defines a set S in terms of a property that its members must satisfy. In this case, the property is that the members (denoted by x) must be greater than 0 and less than or equal to 3.
02

Specifying the Interval

The inequality given, 0 < x 鈮� 3, describes an interval on the number line. Specifically, it represents all numbers greater than 0 and up to and including 3. This interval is open at 0 (because of the strict inequality <) and closed at 3 (because of the inclusive inequality 鈮�).
03

Writing the Interval Notation

The interval corresponding to the set S can be expressed in interval notation as (0, 3]. This notation shows that the interval starts just after 0 and includes 3, matching the inequality previously described.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is an efficient way to describe a range of numbers on the number line. It uses parentheses \( ( ) \) and brackets \( [ ] \) to indicate if the endpoints are included or not. A parenthesis means the endpoint is not included, known as an open interval, while a bracket means the endpoint is included, known as a closed interval.

For instance, in the exercise \( S=\{x: 0
Number Line
A number line is a visual representation of numbers laid out on a straight line, typically with zero at the center, negative numbers to the left, and positive numbers to the right. It's a powerful tool for understanding and solving math problems involving inequalities, such as the one in the set builder notation from the exercise.

On the number line, you can represent an interval by highlighting or shading the region between the indicated numbers. In our example \( S=\{x: 0
Inequalities
Inequalities are mathematical expressions involving the symbols \( < \) (less than), \( > \) (greater than), \( \leqslant \) (less than or equal to), or \( \geqslant \) (greater than or equal to). They are used to show that one quantity is larger or smaller than another or to specify a range of possible values.

In the context of our exercise \( S=\{x: 0

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Most popular questions from this chapter

A function \(f\) defined on an interval \(I=\\{x: a \leqslant x \leqslant b\\}\) is called increasing \(\Leftrightarrow\) \(f\left(x_{1}\right)>f\left(x_{2}\right)\) whenever \(x_{1}>x_{2}\) where \(x_{1}, x_{2} \in I\). Suppose that \(f\) has the intermediate-value property: that is, for each number \(c\) between \(f(a)\) and \(f(b)\) there is an \(x_{0} \in I\) such that \(f\left(x_{0}\right)=c\). Show that a function \(f\) which is increasing and has the intermediate-value property must be continuous.

\(x_{n}=1+\left((-1)^{n} / n\right)\)

Suppose that \(f\) is a continuous, increasing function on a closed interval \(I=\) \(\\{x: a \leqslant x \leqslant b\\}\). Show that the range of \(f\) is the interval \([f(a), f(b)]\)

The sequence $$ 1 \frac{1}{2}, 2 \frac{1}{2}, 3 \frac{1}{2}, 1 \frac{1}{3}, 2 \frac{1}{3}, 3 \frac{1}{3}, 1 \frac{1}{4}, 2 \frac{1}{4}, 3 \frac{1}{4}, \ldots $$ has subsequences which converge to the numbers 1,2 , and 3 . (a) Write a sequence which has subsequences which converge to \(N\) different numbers where \(N\) is any positive integer. (*b) Write a sequence which has subsequences which converge to infinitely many different numbers.

Given the function $$ f: x \rightarrow \begin{cases}1 & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational }\end{cases} $$ (a) Show that \(f\) is not continuous at any \(x_{0}\). (b) If \(g\) is a function with domain all of \(\mathbb{R}^{1}\), if \(g(x)=1\) if \(x\) is rational, and if \(g\) is continuous for all \(x\), show that \(g(x) \equiv 1\) for \(x \in \mathbb{R}^{1}\).
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