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Chapter 13: Problem 3

Write a verbal description of the set \\{3,6,9\\}

Short Answer

Expert verified
The set \{3,6,9\} is described as the first three multiples of 3.

Step by step solution

01

Identify the Pattern

Observe the numbers in the set \(\{3, 6, 9\}\). Each number is a multiple of 3. This indicates a sequence where each term is obtained by multiplying an integer with 3.
02

Determine the Sequence

Notice that 3 is the first term, 6 is the second (3 times 2), and 9 is the third (3 times 3). This suggests that the numbers can be described using the pattern: 'three times an integer starting from 1.'
03

Write the Verbal Description

Combine the patterns identified into a coherent sentence. The numbers can be verbally described as the set of the first three multiples of 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Theory
Set theory is a fundamental aspect of mathematics that deals with the collection of objects, called elements or members, which can be anything from numbers to letters or other abstract concepts. In mathematics, a set is often denoted by curly braces, like \( \{ \} \). Each element inside the braces is separated by a comma. For example, the set \( \{3, 6, 9\} \) is a simple example consisting of numbers.Understanding sets involves recognizing certain properties:
  • **Well-defined nature**: A set is clearly defined by its elements. For instance, any set representing numbers that are multiples of 3 can only contain numbers fitting that definition.
  • **Uniqueness**: Each element in a set is unique. In set theory, \( \{3, 6, 9\} \) will remain the same regardless of order, so \( \{9, 3, 6\} \) is equivalent to \( \{3, 6, 9\} \).
  • **Finite and Infinite Sets**: Sets can be finite, like \( \{3, 6, 9\} \), or infinite, like the set of all multiples of 3.
This makes sets useful to organize mathematical information in a clear and concise way.
Number Sequences
A number sequence is a list of numbers in a specific order that follows a particular rule or pattern. The concept of sequences is that each number in the sequence, called a term, is derived using a specific rule or formula.When dealing with sequences, some basic characteristics include:
  • **Starting Point**: The first number in the sequence is known as the starting point or the first term.
  • **Pattern or Rule**: Each subsequent number is obtained by applying a uniform rule or formula, like adding a specific number, multiplying, or any other operation.
  • **Position of Terms**: Each term can be described by its position. In the sequence \( \{3, 6, 9\} \), 3 is the first term, 6 is the second term, and 9 is the third term.
A clear understanding of the rule governing the sequence allows you to predict future terms, giving insight into patterns within numbers. For the set \( \{3, 6, 9\} \), each term is 3 times an increasing integer.
Multiples
Multiples of a number are formed by multiplying that number by integers. They play a crucial role in understanding patterns within math, such as sequences and sets.Here鈥檚 what you need to know about multiples:
  • **Definition**: A multiple of a number is the product of that number and any integer. For example, multiples of 3 include \(3 \times 1 = 3\), \(3 \times 2 = 6\), \(3 \times 3 = 9\), and so on.
  • **Properties of Multiples**: Multiples of a number continue infinitely as you multiply by larger and larger integers.
  • **Real-World Relevance**: Understanding multiples helps solve problems involving divisibility and finding common denominators.
In the example of the set \( \{3, 6, 9\} \), each element is a multiple of 3. This demonstrates the concept of using multiples to form a set and develop patterns seen in number sequences.

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