12) a. Find the value of n (A) if n ( AU B) = 50, n(B)= 30, and n(A N B) = 10. b. Use Venn diagram to answer the following two questions- Fan response to singers : Julie Davis, a pop culture analysts, wanted to evalute the realtive appeals of different singers, She interviewed 65 fans and determinded the following:
12) a. Find the value of n (A) if n ( AU B) = 50, n(B)= 30, and n(A N B) = 10. b. Use Venn diagram to answer the following two questions- Fan response to singers : Julie Davis, a pop culture analysts, wanted to evalute the realtive appeals of different singers, She interviewed 65 fans and determinded the following:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
It's a two part question. Thank You!
Transcribed Image Text:**Text for Educational Website:**
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**Question: How many of these fans like:**
- **(a)** Exactly two of these singers?
- **(b)** Exactly one of these singers?
- **(c)** None of these singers?
- **(d)** Jazmine, but neither Carrie nor Brad?
- **(e)** Brad and exactly one of the other two?
---
This exercise prompts learners to analyze a set of options regarding music preferences. Consider exploring logical reasoning and set theory to determine the number of fans corresponding to each scenario.
![**Problem 12:**
**a.** Calculate the value of n(A) given the following:
- n(A ∪ B) = 50
- n(B) = 30
- n(A ∩ B) = 10
Use the principle of inclusion-exclusion for sets:
\[ n(A ∪ B) = n(A) + n(B) - n(A ∩ B) \]
Substitute the given values:
\[ 50 = n(A) + 30 - 10 \]
\[ 50 = n(A) + 20 \]
\[ n(A) = 30 \]
**b.** Venn Diagram Analysis:
Julie Davis, a pop culture analyst, surveyed 65 fans to assess the popularity of different singers. The results are as follows:
- 37 fans like Jazmine Sullivan
- 36 fans like Carrie Underwood
- 31 fans like Brad Paisley
- 14 fans like both Jazmine and Carrie
- 21 fans like both Jazmine and Brad
- 14 fans like both Carrie and Brad
- 8 fans like all three singers
**Venn Diagram Explanation:**
- **Jazmine Sullivan Circle:**
- Total: 37 fans
- Overlap with Carrie: 14 fans
- Overlap with Brad: 21 fans
- All three singers: 8 fans
- **Carrie Underwood Circle:**
- Total: 36 fans
- Overlap with Jazmine: 14 fans
- Overlap with Brad: 14 fans
- All three singers: 8 fans
- **Brad Paisley Circle:**
- Total: 31 fans
- Overlap with Jazmine: 21 fans
- Overlap with Carrie: 14 fans
- All three singers: 8 fans
This information allows the construction of a Venn diagram to visually represent the intersections and unique preferences of the surveyed fans.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F119977c1-3a9f-4ed7-9f2b-cc635011c7d9%2F6c65e274-5a62-43ff-925d-4758c5d04c84%2F9zla4s8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 12:**
**a.** Calculate the value of n(A) given the following:
- n(A ∪ B) = 50
- n(B) = 30
- n(A ∩ B) = 10
Use the principle of inclusion-exclusion for sets:
\[ n(A ∪ B) = n(A) + n(B) - n(A ∩ B) \]
Substitute the given values:
\[ 50 = n(A) + 30 - 10 \]
\[ 50 = n(A) + 20 \]
\[ n(A) = 30 \]
**b.** Venn Diagram Analysis:
Julie Davis, a pop culture analyst, surveyed 65 fans to assess the popularity of different singers. The results are as follows:
- 37 fans like Jazmine Sullivan
- 36 fans like Carrie Underwood
- 31 fans like Brad Paisley
- 14 fans like both Jazmine and Carrie
- 21 fans like both Jazmine and Brad
- 14 fans like both Carrie and Brad
- 8 fans like all three singers
**Venn Diagram Explanation:**
- **Jazmine Sullivan Circle:**
- Total: 37 fans
- Overlap with Carrie: 14 fans
- Overlap with Brad: 21 fans
- All three singers: 8 fans
- **Carrie Underwood Circle:**
- Total: 36 fans
- Overlap with Jazmine: 14 fans
- Overlap with Brad: 14 fans
- All three singers: 8 fans
- **Brad Paisley Circle:**
- Total: 31 fans
- Overlap with Jazmine: 21 fans
- Overlap with Carrie: 14 fans
- All three singers: 8 fans
This information allows the construction of a Venn diagram to visually represent the intersections and unique preferences of the surveyed fans.
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