Question Use set-builder notation to express the set of all positive integers that are five times the value of integers greater than 1 and less than 20 . Select the correct answer below: \( \{5 p \mid p \in \mathbb{Z} \) and \( 1<5 p<20\} \) \( \{5 p \mid p \in \mathbb{Z} \) and \( 1<5 p<100\} \) \( \{5 p \mid p \in \mathbb{Z} \) and \( -1

Ask by Newton Carter. in the United States

Mar 09,2025

To solve this problem, we first identify the integers \( p \) that are greater than 1 and less than 20. The integers in consideration are \( p = 2, 3, ..., 19 \). Therefore, we are looking for the set of positive integers \( 5p \) where \( p \) meets these criteria. Now, let's put it into set-builder notation: The correct expression would be \( \{5 p \mid p \in \mathbb{Z}^+ \text{ with } 1 < p < 20 \} \). This means \( p \) must be a positive integer (hence the \( \mathbb{Z}^+ \)) that is between 1 and 20. Now, none of the provided options seem to fit this exact requirement, but we can clarify which is closest in intent. The best fit from the choices would be: \( \{5 p \mid p \in \mathbb{Z} \text{ and } 1< p < 20\} \). Historical Background: Did you know that set-builder notation has its roots in the early 20th century, developed by mathematicians like Ernst Zermelo? It simplifies the description of sets, allowing mathematicians to express large sets without listing out every element. This abstraction is vital for modern mathematics! Real-World Application: Set-builder notation isn't just for classroom exercises—it has real-world applications too! It's used in computer science for database queries and algorithms, and in statistics to define specific data sets. For example, you might define a set of even integers within a certain range when programming a statistical analysis function!