Problem 18 Use set-builder notation to desc... [FREE SOLUTION]

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Chapter 0: Problem 18

Use set-builder notation to describe each set. $$ \\{\ldots,-6,-3,0,3,6, \ldots\\} $$

Short Answer

Expert verified

The set in set-builder notation is $\{ x \; | \; x = 3n, \; n \in \mathbb{Z} \}$.

Step by step solution

- Observe the Pattern

Recognize the pattern in the given set. Notice that the numbers are multiples of 3: $..., -6, -3, 0, 3, 6, ...$

- Introduce Variable

Let any number in the set be represented by the variable $n$. Each element can be written as $3n$, where $n$ is an integer.

- Define the Set in Set-Builder Notation

Use set-builder notation to define the set: \ $\{ x \; | \; x = 3n, \; n \in \mathbb{Z} \}$

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Key Concepts

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

Integer multiples

In mathematics, an integer multiple of a number is formed by multiplying that number by any integer. In the given set, the numbers are examples of integer multiples of 3. For instance, when we multiply 3 by -2, -1, 0, 1, and 2, we get -6, -3, 0, 3, and 6, respectively. This pattern will continue infinitely in both directions. Understanding integer multiples helps in recognizing sequences and sets. A set of integer multiples can be represented compactly using set-builder notation.

Mathematical patterns

Observing patterns in numbers is crucial in mathematics. Here, the given set follows a clear pattern of increasing or decreasing by 3. Recognizing this pattern helps in defining the general form of any number in the set. Such patterns are not only useful for solving problems but also for understanding deeper mathematical concepts and functions. Identifying patterns can simplify complex problems and make them more manageable.

Variable representation

Using variables to represent numbers in a set is a common mathematical practice. In this case, we use the variable 'n' to represent any integer. Each element of the set can be expressed as 3n, where n is any integer. By introducing a variable, we generalize the pattern and provide a concise way to describe all elements in the set. This generalization is then formalized using set-builder notation, which states: \text{${ x \big| x = 3n, n \boldsymbol{\textrm{鈭坿} \boldsymbol{\textrm{鈩}}$}.}Here, \textrm{3n} represents the multiple of 3, and \textrm{$饾憶饾憱饾憶 \bh{饾挼}$} signifies that 'n' can be any integer.

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91影视

Use set-builder notation to describe each set. $$ \\{\ldots,-6,-3,0,3,6, \ldots\\} $$

Short Answer

Step by step solution

- Observe the Pattern

- Introduce Variable

- Define the Set in Set-Builder Notation

Key Concepts

Integer multiples

Mathematical patterns

Variable representation

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