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Chapter 6: Problem 4

Write the set in set-builder notation. $$ \\{1,3,5,7,9,11, \ldots, 39\\} $$

Short Answer

Expert verified
\[\{2n - 1 \, | \, n \in \mathbb{N}, 1 \leq n \leq 20 \}\]

Step by step solution

01

Identify the base set

The base set contains all possible values of the set elements. In this case, it is the set of all natural numbers, denoted by \(\mathbb{N}\).
02

Identify the pattern

The set consists of odd numbers. Odd numbers can be represented as \(2n-1\) where \(n\) is an element of the base set \(\mathbb{N}\).
03

Identify the conditions or boundaries

We want to include only the odd numbers from 1 to 39. The smallest value of the set is 1, and the largest value is 39. We need to find the corresponding values of \(n\) for which \(2n-1\) equal to these numbers. For the smallest value (1), we can set up an equation: \(2n-1=1\). Solving for \(n\), we get: \(n=1\). For the largest value (39), we can set up another equation: \(2n-1=39\). Solving for \(n\), we get: \(n=20\). So, we want to include all odd numbers formed by \(2n-1\) where \(n\) ranges from 1 to 20.
04

Write the set in set-builder notation

Using our analysis, we can now write the given set in set-builder notation: \[\{2n - 1 \, | \, n \in \mathbb{N}, 1 \leq n \leq 20 \}\] This notation means that the set contains elements in the form of \(2n - 1\) with \(n\) being a natural number between 1 and 20, inclusive.

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