Problem 27 - 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. \(I\) _________ \(N\)

Step-by-Step Solution

Answer

\(I \nsubseteq N\) because integers include numbers not in naturals.

Step 1: Understanding the sets

First, we need to understand the types of numbers each set represents. Set \(N\) consists of natural numbers (1, 2, 3, ...), while set \(I\) consists of integers (..., -3, -2, -1, 0, 1, 2, 3, ...). Thus, \(I\) includes negative numbers, zero, and positive numbers.

Step 2: Comparing the sets

Next, compare the inclusiveness of the sets. The natural numbers \(N\) are a subset of integers \(I\), but not all integers \(I\) are natural numbers because the integers include negative numbers and zero, which are not in \(N\).

Step 3: Determining the relationship

Based on our comparison, since not all elements of set \(I\) are elements in set \(N\), set \(I\) is not a subset of \(N\). Therefore, the correct relationship is \(I subseteq N\).

Key Concepts

Natural Numbers

Natural numbers form the basis of number systems by including the simplest form of positive numbers. They are the numbers we typically start counting from, which include 1, 2, 3, and so on.

• Natural numbers are denoted by the set symbol \( N \).
• This set does not contain zero or any negative numbers.
• The representation of these numbers continues indefinitely in the positive direction.

To better understand, think of natural numbers as the numbers used for counting items or referencing order, such as when you are talking about the first, second, third, etc.
Natural numbers are foundational because they are the simplest non-negative numbers that serve as building blocks for other systems.

Integers

Integers expand upon the natural numbers by including zero and the negative numbers that extend indefinitely in the negative direction. This set encompasses every whole number, both positive and negative.

• Integers are denoted by the set symbol \( I \).
• They include natural numbers, zero, and negative counterparts like -1, -2, -3, etc.
• Integers are crucial for operations that require values below zero, presenting a balanced system symmetrical around zero.

The incorporation of integers is essential for mathematical operations involving losses, debts, or temperatures below zero. This symmetry around zero allows for a broader range of calculations and concepts.

Subsets

The concept of subsets helps us understand how different sets of numbers relate to each other. A subset is a set where all of its elements are contained within another set.

• If every element of a set \( A \) is also in set \( B \), then \( A \) is a subset of \( B \), denoted as \( A \subseteq B \).
• If there is even one element in \( A \) that is not in \( B \), then \( A \) is not considered a subset of \( B \), denoted as \( A subseteq B \).
• This concept is critical in defining and understanding the relationships between different number systems, such as how natural numbers relate to integers and other number types.

In the given exercise, the set of integers \( I \) is not a subset of the natural numbers \( N \), because \( I \) contains negative numbers and zero, which are not present in \( N \). Hence, the correct symbol to use in the blank is \( subseteq \). Subsets are crucial for understanding how different sets interlink and extend, providing clarity on inclusivity and exclusivity within sets.