Problem 28 - 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. \(I\)_____ \(W\)

Step-by-Step Solution

Answer

I \nsubseteq W

Step 1: Understand Set Definitions

Let's clarify the set definitions. The set I represents integers, which include whole positive, negative numbers, and zero. The set W represents whole numbers, which are non-negative integers (zero included).

Step 2: Determine the Relationship

We need to consider whether set I, which includes both positive and negative integers as well as zero, is entirely contained in set W, which only includes non-negative integers and zero. Set W lacks negative integers, which are present in set I.

Step 3: Conclusion About the Subset Relation

Since set I contains elements like -1, -2, etc., which are not in set W, I cannot be a subset of W. Therefore, I is not entirely contained within W.

Key Concepts

Natural Numbers

Natural numbers are the foundation of the number systems used in mathematics. They are the simplest set of numbers that we encounter in mathematics and daily life. Natural numbers typically start from 1 and go upward: 1, 2, 3, 4, and so on. They represent whole, positive numbers and are often used for counting objects.

• Natural numbers do not include zero in the classical sense, although some modern definitions might include zero.
• Natural numbers do not include any fractions, decimals, or negative numbers.
• They are denoted by the symbol \( \mathbb{N} \).

Understanding natural numbers helps form the basic building blocks for other sets of numbers, such as integers and real numbers, which have additional characteristics and include more complex values.

Integer Numbers

Integer numbers expand upon the natural numbers by including negative numbers and zero. They form a broader set that still does not encompass fractions or decimals. Integers can be thought of as the set of whole numbers, both positive and negative.

• The set of integers is represented as \( \mathbb{Z} \).
• Integers include numbers like -3, -2, -1, 0, 1, 2, 3, and so forth.
• This set includes all natural numbers as well, since all natural numbers are non-negative integers.

Integers are crucial when considering mathematical operations like subtraction, where the result may not always be a natural number. Additionally, being a broader set allows integers to form a complete number line, where every whole point on this line represents an integer.

Real Numbers

Real numbers are an even larger set that includes all the possible magnitudes of numbers we encounter in mathematics. They encompass both rational and irrational numbers, providing a complete system for analyzing virtually all types of values.

• Real numbers include natural numbers, whole numbers, integers, rational numbers (like fractions), and irrational numbers (like \(\pi\) and \(\sqrt{2}\)).
• The set of real numbers is symbolized by \( \mathbb{R} \).
• Real numbers cover everything on the number line, with no gaps.

This set is essential for continuous mathematics, as it allows for precise measurement and computation over a vast array of values.

Subset Relation

In set theory, the concept of a subset is pivotal in understanding the relationship between different sets. A set \( A \) is a subset of set \( B \) if every element of \( A \) is also an element of \( B \). This is denoted by \( A \subseteq B \).

• If at least one element of \( A \) is not in \( B \), then \( A \) is not a subset of \( B \), indicated by \( A subseteq B \).
• Determining subset relations requires comparing each element of one set to see if it is part of another.
• For example, the integers \( \mathbb{Z} \) include negative numbers, which are not included in the whole numbers \( \mathbb{W} \). Thus, \( \mathbb{Z} subseteq \mathbb{W} \).

Understanding these relationships allows us to better work with different families of numbers and understand how they interact with one another in problems involving set theory.