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Analyzing Toothpaste Flavor Preferences Using Set Theory

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In this example, we explore the flavor preferences of toothpaste among a sample of 120 people, considering regular and mint flavors. By setting up a Venn diagram, we can visually analyze the relationships between the groups: those who liked only regular, only mint, both flavors, and neither. The calculations reveal that 39 people preferred only regular, 27 preferred only mint, and 19 did not like either flavor. This application of sets and Venn diagrams illustrates how to effectively analyze survey data in decision-making contexts, such as product development.

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Analyzing Toothpaste Flavor Preferences Using Set Theory

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  1. AND

  2. Chapter 2 Sets

  3. • Application of sets WHAT YOU WILL LEARN

  4. Section 5 Applications of Sets

  5. Example: Toothpaste Taste Test • A drug company is considering manufacturing a new toothpaste. They are considering two flavors, regular and mint. • In a sample of 120 people, it was found that 74 liked the regular, 62 liked the mint, and 35 liked both types. • How many liked only the regular flavor? • How many liked either one or the other or both? • How many people did not like either flavor?

  6. Begin by setting up a Venn diagram with sets A (regular flavor) and B (mint flavor). Since some people liked both flavors, the sets will overlap and the number who liked both with be placed in region II. 35 people liked both flavors. U 35 II A(Regular) B(Mint) Solution

  7. Next, region I will refer to those who liked only the regular and region III will refer to those who liked only the mint. In order to get the number of people in each region, find the difference between all the people who liked each toothpaste and those who liked both. I: 74 – 35 = 39 III: 62 – 35 = 27 U 39 regular only 27 mint only A(Regular) B(Mint) Solution (continued) 35 both II I III

  8. Solution (continued) • “One or the other or both” represents the UNION of the two sets. Therefore, 39 + 27 + 35 = 101 people who liked one or the other or both.

  9. 19 liked U ne i ther 35 74 - 35 = 3 9 62 - 35 = 27 both L iked regular L iked mint only only A(Regular) B(Mint) Solution (continued) • Take the total number of people in the entire sample and subtract the number who liked one or the other or both. 120-101 = 19 people did not like either flavor. I I I I I I

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