Problem 35 - 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 whole number less than 6\(\\}\)

Step-by-Step Solution

Answer

The elements of the set are \( \{0, 1, 2, 3, 4, 5\} \).

Step 1: Understanding the Set Builder Notation

The given set is written using set builder notation: \( \{ n \mid n \text{ is a whole number less than 6} \} \). This notation defines the set by describing the properties that its elements must satisfy. Specifically, it means the set of all numbers \( n \) that are whole numbers and less than 6.

Step 2: Defining Whole Numbers

Whole numbers are non-negative numbers without fractions or decimals. They start from 0 and include 0, 1, 2, 3, and so on. For this exercise, we need to determine which whole numbers satisfy the condition of being less than 6.

Step 3: Identifying the Elements

List all whole numbers starting from 0 up to those that are less than 6. These are \( 0, 1, 2, 3, 4, \) and \( 5 \). Since 6 itself is not less than 6, it is not included.

Step 4: Writing the Set

Compile the identified elements into a set. The set of all whole numbers less than 6 is \( \{0, 1, 2, 3, 4, 5\} \). This set contains all the whole numbers that meet the criteria outlined in the set builder notation.

Key Concepts

Whole Numbers

Whole numbers are an essential concept in mathematics, forming a subset of the number system. They are non-negative integers that do not contain fractions, decimals, or negative signs. This set begins from 0 and moves upward infinitely in a linear sequence:

• 0
• 1
• 2
• 3
• 4
• ...

Whole numbers are very practical in counting and often used in set theory for establishing elements of a set. To determine if a number is a whole number, simply check: - it should be a non-negative integer - should not include fractions or decimals.
For instance, in a set defined by set builder notation as containing numbers less than a specified limit, whole numbers are often involved just like in the example where every number below 6 is considered.

Set Theory

Set theory is a fundamental area of mathematics dealing with collections of objects called sets. Sets can include anything: numbers, people, letters, etc. For our purposes, they often include numbers like whole numbers. Set theory helps us understand how to group different, distinct objects together.
In mathematical terms, a set is simply a collection of distinct objects or numbers, which are referred to as elements. In set builder notation, a set is defined by specific properties that its elements must satisfy. The notation follows this basic structure:

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

This refers to the set of all elements \(x\) that satisfy specified properties.
For example, in the set \(\{n \mid n\text{ is a whole number less than 6}\}\), we define the elements as the whole numbers that are less than 6.

Elements of a Set

The elements of a set are the individual objects or numbers contained within the set. An element's role in a set is determined by the rules or properties defined during the set's creation. Each element in a set is unique, meaning it cannot repeat.
Let's take the example of examining the set \(\{n \mid n \text{ is a whole number less than 6}\}\). Here, the elements of the set are the whole numbers that meet the condition of being less than 6. These numbers include: 0, 1, 2, 3, 4, and 5.
Using set builder notation ensures that we only list elements that adhere to the predefined rules. This allows for precise and accurate representation of sets in mathematics, which is crucial for solving problems and understanding more complex mathematical concepts.