Problem 22 - 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. \(N\)_____ \(I\)

Step-by-Step Solution

Answer

\(N \subseteq I\) because every natural number is an integer.

Step 1: Understand the Definitions

We need to understand the definitions of the sets involved:- The set of natural numbers, \(N\), includes positive integers \(\{1, 2, 3, \ldots\}\).- The set of integers, \(I\), includes all whole numbers and their negatives \(\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\).

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

\(N \subseteq I\) because every natural number is an integer.

Key Concepts

Natural Numbers

Natural numbers are the numbers we often start counting with. These are the numbers we use when counting objects or people and start from 1. Therefore, natural numbers include 1, 2, 3, and so on, extending infinitely in the positive direction. As they are used in everyday counting, they do not include zero, negative numbers, fractions or decimals.

• Natural numbers are represented by the symbol \( N \).
• The sequence is infinite and only consists of positive whole numbers.
• They are sometimes referred to as "counting numbers" because that's their primary use.

Understanding natural numbers is essential as they serve as the foundation for building other number sets, such as integers and whole numbers.

Integers

Integers expand on the concept of natural numbers by including negative numbers and zero. Essentially, integers cover the entire spectrum of whole numbers that stretch infinitely in both the positive and negative directions. This means that while natural numbers are a part of integers, integers include additional numbers.

• Integers are represented by the symbol \( I \).
• The set includes numbers like \( \{-3, -2, -1, 0, 1, 2, 3, \ldots\} \).
• Integers are critical for calculations that involve negative values, which can occur in various real-world scenarios like calculations of debt, temperature below zero, etc.

The ability to recognize and use integers helps to understand more complex aspects of mathematics.

Subset Relation

In set theory, a subset is a set whose elements are all contained within another set. The subset relation is denoted by the symbol \( \subseteq \). This means if all members of set \( A \) are also members of set \( B \), then \( A \subseteq B \).
For the set of natural numbers \( N \) and the set of integers \( I \), the relationship \( N \subseteq I \) holds true. This is because every natural number is also an integer. However, not all integers are natural numbers since integers include zero and negative numbers, which are not part of the natural numbers.

• When understanding subset relationships, observe if every element of one set belongs in another set absolutely.
• If a set contains all elements plus additional ones, the initial set is a subset of this larger set.

The subset relation is important in mathematics for establishing hierarchies and relationships between different sets of numbers.