Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Problem Statement:
Given the following Venn diagram, find \( n[(B \cap C)^c] \).
### Explanation of the Venn Diagram:
The diagram consists of three intersecting circles labeled A, B, and C, representing different sets. Each section of the diagram contains numbers representing the number of elements in each specific subset. The universal set U is represented by a rectangle containing all the sets.
- **Set A:**
- Only A: 23
- A and B but not C: 11
- A and C but not B: 6
- A, B, and C: 14
- **Set B:**
- Only B: 9
- A and B but not C: 11
- B and C but not A: 15
- A, B, and C: 14
- **Set C:**
- Only C: 21
- A and C but not B: 6
- B and C but not A: 15
- A, B, and C: 14
- **Outside sets A, B, C (In Universal Set U):** 11
### Question:
Find the number of elements in the complement of the intersection of sets B and C, \( n[(B \cap C)^c] \).
### Options:
- 15
- 81
- 25
- 29
- 85
- None of the above.
---
To solve the problem, determine the elements in \( B \cap C \):
- \( B \cap C \) includes the regions:
- B and C but not A: 15
- A, B, and C: 14
So, \( n(B \cap C) = 15 + 14 = 29 \).
Find the complement of \( B \cap C \):
- Total in Universal Set U = Sum of all numbers = \( 23 + 11 + 6 + 14 + 9 + 15 + 21 + 11 \)
- Calculating \( n(U) \): \( 110 \).
- Therefore, \( n[(B \cap C)^c] = n(U) - n(B \cap C) = 110 - 29 = 81 \).
Hence, the correct answer is 81.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5fed828e-39c5-45a6-bd50-5b2345ce7f7e%2Fb166768e-3e9e-4db9-bbaf-6022f09f4d34%2Fm1z5emo_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Given the following Venn diagram, find \( n[(B \cap C)^c] \).
### Explanation of the Venn Diagram:
The diagram consists of three intersecting circles labeled A, B, and C, representing different sets. Each section of the diagram contains numbers representing the number of elements in each specific subset. The universal set U is represented by a rectangle containing all the sets.
- **Set A:**
- Only A: 23
- A and B but not C: 11
- A and C but not B: 6
- A, B, and C: 14
- **Set B:**
- Only B: 9
- A and B but not C: 11
- B and C but not A: 15
- A, B, and C: 14
- **Set C:**
- Only C: 21
- A and C but not B: 6
- B and C but not A: 15
- A, B, and C: 14
- **Outside sets A, B, C (In Universal Set U):** 11
### Question:
Find the number of elements in the complement of the intersection of sets B and C, \( n[(B \cap C)^c] \).
### Options:
- 15
- 81
- 25
- 29
- 85
- None of the above.
---
To solve the problem, determine the elements in \( B \cap C \):
- \( B \cap C \) includes the regions:
- B and C but not A: 15
- A, B, and C: 14
So, \( n(B \cap C) = 15 + 14 = 29 \).
Find the complement of \( B \cap C \):
- Total in Universal Set U = Sum of all numbers = \( 23 + 11 + 6 + 14 + 9 + 15 + 21 + 11 \)
- Calculating \( n(U) \): \( 110 \).
- Therefore, \( n[(B \cap C)^c] = n(U) - n(B \cap C) = 110 - 29 = 81 \).
Hence, the correct answer is 81.
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