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Chapter 1: Problem 9

Write each set by listing its elements. See Example \(1\). \(\\{x | x \text { is an irrational number that is also rational }\\}\)

Short Answer

Expert verified
The set is empty and can be denoted by \({\emptyset}\).

Step by step solution

01

Understand the Definitions

First, identify what irrational numbers and rational numbers are. Irrational numbers cannot be expressed as a fraction of two integers. Examples include \(\sqrt{2}\), \(\pi\), and so on. Rational numbers can be written as a fraction where both numerator and denominator are integers, with denominator not equal to zero.
02

Analyze the Condition

The set described includes numbers that are both irrational and rational at the same time. Consider whether such numbers exist.
03

Conclude the Set Elements

Since a number cannot be both rational and irrational simultaneously, there are no elements that satisfy this condition. Therefore, the set is empty.
04

Write the Set

List the elements of the set. Since the set is empty, denote it using empty set notation.

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Key Concepts

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

irrational numbers
Irrational numbers play an important role in intermediate algebra. Unlike rational numbers, irrational numbers cannot be written as simple fractions. They cannot be expressed as a ratio of two integers. Examples of irrational numbers include \( \sqrt{2} \), \( \pi \), and \( \sqrt{3} \).

These numbers have non-repeating and non-terminating decimal representations. This means their decimal expansions go on forever without repeating.

If we say a number is irrational, it means we cannot write it in the form \( \frac{p}{q} \), where both \( p \) and \( q \) are integers, and \( q eq 0 \).
rational numbers
Rational numbers are much simpler to understand. They are numbers that can be expressed as the ratio of two integers. This means any number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \), is a rational number.

Examples include:
  • \( \frac{1}{2} \)
  • \( \frac{-3}{4} \)
  • \( 5 \) (since 5 can be written as \( \frac{5}{1} \))

Rational numbers have decimal representations that terminate (like 0.75) or repeat (like 0.333...).

Understanding the difference between rational and irrational numbers is key to solving many algebra problems.
empty set
The empty set is a fundamental concept in mathematics. It is defined as a set that has no elements. In other words, it is 'empty'. We denote the empty set using the symbol \( \emptyset \) or simply \( \{ \} \).

In the example problem, we were asked to find numbers that are both irrational and rational at the same time. Such numbers do not exist because a number cannot be both at once. Thus, the set of such numbers is empty.

This concept helps us understand impossible conditions. If a set contains no elements meeting given criteria, it is the empty set. Understanding this concept is crucial for solving various algebraic problems.

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Most popular questions from this chapter

Complete each statement and give an example. The product of two numbers with different signs is ______.

Complete each statement and give an example. The difference between two positive numbers is negative if _______

Which elements of each set are (a) natural numbers. (b) whole numbers, (c) integers, (d) rational numbers, (e) irrational numbers, (f) real numbers? See Example 3. $$ \left\\{-8,-\sqrt{5},-0.6,0, \frac{3}{4}, \sqrt{3}, \pi, 5, \frac{13}{2}, 17, \frac{40}{2}\right\\} $$

Graph the elements of each set on a number line. $$ \left\\{-\frac{6}{5},-\frac{1}{4}, 0, \frac{5}{6}, \frac{13}{4}, 5.2, \frac{11}{2}\right\\} $$

Write each set by listing its elements. See Example \(1\). \(\\{k | k \text { is an odd integer less than } 1\\}\)
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