Problem 41 - Algebra for College Students

Question

List the elements of each set. For example, the elements of \(\\{x \mid x\) is a natural number less than 4\(\\}\) can be listed as \(\\{1,2,3\\}\). \(\\{n \mid n\) is a nonnegative integer less than 5\(\\}\)

Step-by-Step Solution

Answer

\(\{0, 1, 2, 3, 4\}\)

Step 1: Understand the Set Notation

The set is given in set-builder notation: \( \{n \mid n \text{ is a nonnegative integer less than 5}\} \). This means we need to list all nonnegative integers that are less than 5.

Step 2: Identify Nonnegative Integers

Nonnegative integers are numbers from the set of whole numbers that include 0. For our purposes, these numbers are 0, 1, 2, 3, 4, and so on.

Step 3: Apply the Condition

According to the condition, \(n \) must be less than 5. Therefore, we only consider nonnegative integers that are below 5: 0, 1, 2, 3, 4.

Step 4: List the Elements of the Set

The elements that satisfy the condition are 0, 1, 2, 3, and 4. List these numbers in curly brackets to represent the set: \( \{0, 1, 2, 3, 4\} \).

Key Concepts

Set-Builder Notation

Set-builder notation is a concise way of describing a set by specifying a property that its members must satisfy. In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
Set-builder notation is commonly written in the form:

• \( \{ x \mid \text{condition involving } x \} \)

This basically reads as "the set of all \(x\) such that \(x\) satisfies the given condition." The vertical bar "\( \mid \)" can be read as "such that." This is an efficient way to describe large or complex sets without listing every element.
Understanding set-builder notation is crucial because it allows for the expression of sets, particularly infinite ones, in a compact and generalized way. It can express conditions such as inequalities, properties, or equations that define the elements of the set.

Nonnegative Integers

Nonnegative integers are the numbers in the set of whole numbers that are greater than or equal to zero. They do not include negative numbers. Nonnegative integers start from 0 and include numbers like 0, 1, 2, 3, and so forth.
These numbers can be represented by the set:

• \( \{0, 1, 2, 3, \ldots\} \)

Nonnegative integers include the familiar numbers we often use in counting and ordering. They are particularly important in processes that do not involve negative or fractional components, such as counting objects or describing certain conditions in algorithms.
When you see the term "nonnegative integer" in set-builder notation, it specifies that you are interested in whole numbers starting from zero.

Listing Elements of a Set

Listing elements of a set involves writing out all the individual members of the set explicitly. This process is particularly straightforward for finite sets where you can easily enumerate all the elements.
To list the elements of a set given in set-builder notation, follow these guidelines:

• Understand the conditions specified in the set-builder notation.
• Identify the range or constraints given, such as "less than a certain number."
• Use the constraints to determine which specific elements fit the description.
• Write these elements in curly braces, separated by commas.

For instance, if you have the set \( \{ n \mid n \text{ is a nonnegative integer less than 5} \} \), you identify that \(n\) must include nonnegative integers less than 5. Upon evaluation, the set you would list is \( \{ 0, 1, 2, 3, 4 \} \). This methodical approach ensures you accurately convey the elements of the set, reflecting their exact inclusivity defined by the conditions.