Problem 21 - Intermediate Algebra

Question

Use the following set designations. \(N=\\{x \mid x\) is a natural number \(\\}\) \(Q=\\{x \mid x\) is a rational number \(\\}\) \(W=\\{x \mid x\) is a whole number \(\\}\) \(H=\\{x \mid x\) is an irrational number \(\\}\) \(I=\\{x \mid x\) is an integer \(\\}\) \(R=\\{x \mid x\) is a real number \(\\}\) Place \(\subseteq\) or \(\nsubseteq\) in each blank to make a true statement. \(I\)__________\(Q\)

Step-by-Step Solution

Answer

\(I \subseteq Q\)

Step 1: Understand Set Relationships

First, recall the definitions of the sets. The set \(I\) represents all integers, while the set \(Q\) includes all rational numbers. A rational number can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\). Every integer can be expressed as a fraction by setting \(b = 1\).

Step 2: Determine Subset or Not

Since any integer can be written as a fraction (the integer itself divided by 1), it means that every integer is a rational number. Therefore, the set of integers \(I\) is a subset of the set of rational numbers \(Q\).

Step 3: Select the Correct Symbol

The correct set relationship between \(I\) and \(Q\) is \(I \subseteq Q\). This indicates that all elements of \(I\) are included in \(Q\).

Key Concepts

Natural Numbers

Natural numbers are one of the most basic sets of numbers used in mathematics. They are often referred to as the "counting numbers" because they are the numbers we use when counting objects, starting from 1. Natural numbers are positive integers and do not include zero or any fractions or decimals. For example:

• 1, 2, 3, 4, 5, ...

These numbers are straightforward and are denoted by the symbol \(N\). They form the foundation for other number sets. Natural numbers are infinite, meaning there's no end to how high they can go. Whenever you think of counting, you're likely thinking in terms of natural numbers. This concept is especially important when learning about basic arithmetic operations like addition and multiplication. Natural numbers are the building blocks of arithmetic and form the basis for understanding more complex sets like integers and rational numbers.

Rational Numbers

Rational numbers are numbers that can be expressed as fractions of two integers, where the denominator is not zero. Formally, a rational number is any number that can be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). Here are some examples:

• \( \frac{1}{2}, \frac{-3}{4}, 5 \left(\text{which can be written as } \frac{5}{1}\right), 0 \left(\text{as } \frac{0}{1}\right)\)

Interestingly, all integers are rational numbers because any integer \( a \) can be expressed as \( \frac{a}{1} \). This concept is crucial when exploring mathematical properties and operations such as division and rational equations. Rational numbers include both positive and negative numbers, and their representation on a number line is dense, which means between any two rational numbers, there is another rational number. Recognizing rational numbers is vital for understanding how they relate to integers and other types of numbers.

Integers

Integers are a set of numbers that include all the positive whole numbers, their negatives, and zero. In simpler terms, integers comprise all the numbers you see on a typical number line without any fractions or decimals. The set of integers is represented by \(I\) and includes:

• ..., -3, -2, -1, 0, 1, 2, 3, ...

Integers are closed under the operations of addition, subtraction, and multiplication, meaning performing these operations on any two integers will result in another integer. When it comes to division, however, that's not always the case as it can lead to rational or even irrational numbers. Understanding integers is important as they are widely used in everyday mathematics, from measuring temperature to evaluating financial datasets. They serve as a bridge between the more basic natural numbers and the broader set of rational numbers.

Real Numbers

Real numbers are perhaps the most comprehensive set of numbers used in everyday mathematics. This set includes all the rational numbers as well as all irrational numbers. Real numbers can be positive, negative, or zero, and can be represented on a number line. Examples of real numbers include:

• Rational numbers like \( \frac{4}{5} \) or 3
• Irrational numbers like \( \sqrt{2} \) or \( \pi \)
• Integers, whole numbers, and zero

Real numbers are denoted by the symbol \(R\), and they are key in understanding a multitude of mathematical concepts and principles. Unlike rational numbers, irrational numbers cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Real numbers are essential in calculus, algebra, and many other areas of mathematics, making them foundational in both theoretical and applied mathematical studies.