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Chapter 0: Problem 15

Use set-builder notation to describe each set. $$ \\{2,4,6,8\\} $$

Short Answer

Expert verified
The set is \(\{2n \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 4\}\).

Step by step solution

01

Identify the Pattern

Examine the given set \(\{2,4,6,8\}\). Notice that these are all even numbers.
02

Determine a General Formula

The given set consists of even numbers starting from 2. An even number can be expressed as \(2n\), where \(n\) is a positive integer.
03

Identify the Range for n

To match the specific elements of the set \(\{2,4,6,8\}\), ensure that \(n\) covers just the numbers needed. Here, \(n = 1,2,3,4\).
04

Construct the Set-Builder Notation

Combine the previous insights to write the set in set-builder notation. The set \(\{2,4,6,8\}\) can be described as \(\{2n \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 4\}\).

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Key Concepts

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

Understanding Even Numbers
Even numbers are integers that can be divided by 2 without a remainder. In other words, if you divide an even number by 2, the result is a whole number. Even numbers are of the form:

\[2, 4, 6, 8, \ldots\] \ An easy way to identify even numbers is to check if they end in 0, 2, 4, 6, or 8.
For example:
  • 2 is even (2 梅 2 = 1)
  • 4 is even (4 梅 2 = 2)
  • 6 is even (6 梅 2 = 3)
  • 8 is even (8 梅 2 = 4)
Understanding this pattern helps us recognize even numbers in a set.
Deriving a General Formula
To describe a series of numbers like \{2,4,6,8\}\, we need a formula that can generate each number.
The sequence in the exercise consists of even numbers, which can be represented by the formula:

\[2n\] \ Here, \ is a positive integer.
This formula means that for \ = 1\, we get
2 \times 1 = 2\,
for \ = 2\, we get \2 \times 2 = 4\, and so forth, creating the sequence.
Essentially, \2n\ will produce any even number when \ is any positive integer.
Establishing the Range for Variables
When dealing with set-building notation, identifying the range of our variable is crucial to accurately express the set.
For the sequence \{2,4,6,8\}\, the numbers are \{2n\}\ with specific values of \.
We need to determine which values of \ will generate our given sequence. As discussed earlier:
  • If \ = 1\, we get 2
  • If \ = 2\, we get 4
  • If \ = 3\, we get 6
  • If \ = 4\, we get 8
So, the variable \ ranges from 1 to 4.
This is represented as: \[1 \leq n \leq 4.\]
When combined, our set-builder notation becomes: \{2n \mid n \in \mathbb{N} \text{ and } 1 \leq n \leq 4\}.\

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