Problem 21 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(I \subseteq Q\) because every integer is a rational number.

Step 1: Understand Subset Relation

To determine if one set is a subset of another, we must see if every element of the first set is also an element of the second set. In other words, if all integers (set \(I\)) can also be classified as rational numbers (set \(Q\)).

Step 2: Examine Set Definitions

Set \(I\) consists of all integers, which include positive numbers, negative numbers, and zero. A rational number (set \(Q\)) is any number that can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\).

Step 3: Determine Integer Representation as Rationals

Any integer \(n\) can be expressed as a fraction \(\frac{n}{1}\). Since the denominator is 1, which is an integer, every integer is a rational number.

Step 4: Conclude Set Relationship

Given that every integer can be expressed as a rational number, it can be concluded that set \(I\) is indeed a subset of set \(Q\). Therefore, \(I \subseteq Q\).

Key Concepts

Integers

Integers are a foundational concept in mathematics that encompass all whole numbers, both positive and negative, and zero. They form an infinite set which can be denoted as \( \{ ..., -3, -2, -1, 0, 1, 2, 3, ... \} \). Every positive integer is also known as a natural number, which starts from 1 and goes to infinity.
What makes integers special is their completeness when it comes to arithmetic operations like addition, subtraction, and multiplication. For instance:

• Addition: 3 + (-2) = 1
• Subtraction: -5 - (-3) = -2
• Multiplication: 4 * (-6) = -24

However, division of integers does not always result in an integer, such as 10 divided by 3, which equals non-integer 3.33. This distinction leads us to the next concept: rational numbers.

Rational Numbers

A rational number is any number that can be expressed as the quotient or fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b eq 0 \). This definition broadens the horizon of numbers to include fractions like \( \frac{3}{4} \) and negative fractions such as \( -\frac{5}{2} \).
Some interesting aspects of rational numbers include:

• They can be whole numbers, which are fractions with a denominator of 1 (like \( \frac{8}{1} = 8 \)).
• They include repeating or terminating decimals. For example, 0.5 is a rational number because it equals \( \frac{1}{2} \).
• Every integer is a rational number since an integer \( n \) can be written as \( \frac{n}{1} \).

Rational numbers are essential in fields that require partitioning or ratios, such as statistics and economics. Their understanding also enhances our grasp on operations and mathematical solutions.

Subsets

In set theory, the concept of subsets is crucial. A set \( A \) is a subset of set \( B \) if every element of \( A \) is also an element of \( B \). Symbolically, this is expressed as \( A \subseteq B \). If this is not the case, we use the symbol \( A subseteq B \).
For a more visual understanding, imagine a subset as a smaller circle within a larger circle, where every point in the smaller circle is shared with the larger one. Let's consider sets:

• Set \( I \), representing integers \( \{..., -2, -1, 0, 1, 2, ...\} \).
• Set \( Q \), representing rational numbers like \( \frac{2}{3}, 4, -1 \).

Since every integer can be expressed as a rational number by being over 1 (like \( 4 = \frac{4}{1} \)), integers are a subset of rational numbers. Therefore, \( I \subseteq Q \). The subset concept allows us to analyze relationships between different number types and understand the structure of number systems.