Problem 35 - Intermediate Algebra

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

\(\{0, 1, 2, 3, 4, 5\}\)

Step 1: Understand Whole Numbers

Whole numbers are non-negative integers. They start from 0 and increase by 1 each time (0, 1, 2, 3,...).

Step 2: Identify the Condition

The condition given is that the whole number must be less than 6. This means n should be smaller than 6.

Step 3: List the Elements

List all whole numbers starting from 0 up to, but not including, 6. These are: 0, 1, 2, 3, 4, 5.

Key Concepts

Whole Numbers

Whole numbers are a key concept in mathematics, representing a set of numbers that are non-negative and without any fractional or decimal part. In simpler terms, whole numbers start from 0 and go upwards to infinity.

• The smallest whole number is 0.
• Whole numbers include numbers like 0, 1, 2, 3, 4, and so on.
• These numbers do not include negative numbers, fractions, or decimals.

In this exercise, understanding that whole numbers less than 6 include 0, 1, 2, 3, 4, and 5 is crucial for listing the correct set.

Listing Elements

When faced with a mathematical set, listing elements clearly and accurately is essential. This involves understanding what values belong to the set and presenting them in a way that is easy to read and comprehend.

• Start by identifying the rule or property that defines your set.
• Arrange the elements based on the property, in this case, whole numbers.
• List them in increasing numerical order to make them clear and unambiguous.

For example, a set of whole numbers less than 6 {0, 1, 2, 3, 4, 5} means writing out each number individually.

Mathematical Notation

Mathematical notation is the language of mathematics used to express concepts clearly and concisely. It uses symbols and standard conventions to avoid ambiguity. Common notations include set notation, which we see in this exercise.

• A set is typically noted by curly braces, such as \( \{ \ldots \} \).
• Rules for including elements are often described inside the set braces, using a vertical bar (\( \mid \)), meaning "such that".
• This means \( \{ n \mid n \text{ is a whole number less than } 6 \} \) translates directly to a list of whole numbers under 6.

Recognizing this notation is important for accurately interpreting and solving such exercises.

Inequality Concepts

Inequalities are mathematical expressions used to compare values. Understanding them helps determine which numbers meet the criteria of a given condition.

• The symbol "less than" (\( < \)) compares two values, showing that one value is smaller than the other.
• In this problem, the inequality \( n < 6 \) indicates that any whole number considering must be less than 6.
• This effectively narrows down the possibilities to 0, 1, 2, 3, 4, and 5.

By understanding inequality symbols, you can interpret the conditions they describe, critical for correctly listing elements in such problems.