Problem 26 - 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. \(W\) ______ \(I\)

Step-by-Step Solution

Answer

\(W \subseteq I\)

Step 1: Understanding Set Relationships

First, let's understand the definitions of the sets involved in the problem. The set of whole numbers, denoted as \(W\), includes all natural numbers and extends to include zero (e.g., \( \{0, 1, 2, 3, ...\} \)). The set of integers, denoted as \(I\), consists of all natural numbers, their negatives, and zero (e.g., \( \{..., -3, -2, -1, 0, 1, 2, 3, ...\} \)). We need to determine if \(W\) is a subset of \(I\).

Step 2: Determining Subset or Not

A set \(A\) is a subset of another set \(B (A \subseteq B)\) if every element of \(A\) is also an element of \(B\). Apply this to our problem: check if every element of \(W\) (whole numbers) is contained within \(I\) (integers). The whole numbers \( \{0, 1, 2, 3, ...\} \) are clearly part of the broader set of integers since integers include negative numbers as well as whole numbers. Thus, \(W \subseteq I\).

Key Concepts

Natural Numbers

Natural numbers are the most basic set of numbers you begin to learn in mathematics. They are the numbers you use when you count objects in day-to-day life. This set includes all the positive numbers from 1 onwards, like 1, 2, 3, 4, and so on. It is an infinite set, which means it goes on forever.

Here are some characteristics of natural numbers:

• The smallest natural number is 1.
• Natural numbers do not include 0.
• They do not include negative numbers, fractions, or decimals.
• They are used for counting or ordering.

Natural numbers are often denoted by the letter \(N\). They form the foundation for other types of numbers you'll learn about.

Whole Numbers

Whole numbers are very similar to natural numbers but with one important addition: the inclusion of 0. These numbers are represented as \(W\), and consist of 0 and all positive numbers like 1, 2, 3, and so forth.

Key points about whole numbers include:

• Whole numbers begin with 0 (zero).
• They do not include negative numbers, fractions, or decimals.
• Whole numbers are ideal for situations where zero is a meaningful number, like in counting completed tasks or items.

The whole numbers set extends the set of natural numbers, making them very useful in different mathematical operations and real-life scenarios.

Integers

Integers expand on whole numbers by including all negative numbers, as well as, 0 and positive numbers. This means the set of integers, denoted \(I\), looks like this: \(",-3, -2, -1, 0, 1, 2, 3, ..." \).

This set has several important properties:

• Integers include both negative and positive whole numbers including zero.
• They are useful in scenarios where something can have a negative quantity, like temperatures below zero or financial debts.
• Just like natural and whole numbers, integers do not include fractions or decimals.

Understanding integers helps in performing operations like addition, subtraction, and understanding concepts like opposite numbers and absolute values.

Subsets

In set theory, a subset is a set where every member is also an element of another set. We use the symbol \( \subseteq \) to denote the subset relationship. For example, if set \(A\) is a subset of set \(B\) (\(A \subseteq B\)), then every element of \(A\) is also found in \(B\).

Important aspects of subsets include:

• If every element of one set is contained within a second set, the first set is a subset of the second.
• A set is always a subset of itself.
• The empty set is a subset of every set.

In the exercise given, determining if \(W\) (whole numbers) is a subset of \(I\) (integers) involves confirming that all elements in \(W\) are also in \(I\). Since this is true, we say \(W \subseteq I\). This concept helps us understand how different number groups relate to each other.