https://file.aiquickdraw.com/mathbot/file/1726151607052pdz04zfa.pdf

The uploaded file contains a range of set theory and Venn diagram problems. Let's go through each one, providing answers and explanations:

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### 1. Set Operations on Universal Set \( U \)

Given:
- \( U = \{1, 2, 3, \dots, 20\} \)
- \( A \), \( B \), and \( C \) are subsets where:
  - \( A \): Divisors of 12 in \( U \)
  - \( B \): Divisors of 15 in \( U \)
  - \( C \): Divisors of 20 in \( U \)

#### (a) \( A \cap B \)
- \( A = \{1, 2, 3, 4, 6, 12\} \)
- \( B = \{1, 3, 5, 15\} \)
- \( A \cap B = \{1, 3\} \)

#### (b) \( B \cap C \)
- \( C = \{1, 2, 4, 5, 10, 20\} \)
- \( B \cap C = \{1, 5\} \)

#### (c) \( A \cap B \cap C \)
- \( A \cap B \cap C = \{1\} \)

#### (d) \( A \cup C \)
- \( A \cup C = \{1, 2, 3, 4, 5, 6, 10, 12, 20\} \)

#### (e) \( (A \cap B) \cup C \)
- \( (A \cap B) = \{1, 3\} \)
- \( (A \cap B) \cup C = \{1, 2, 3, 4, 5, 6, 10, 12, 20\} \)

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### 2. Venn Diagram Problem with Meat Preferences

Given:
- \( n(U) = 350 \)
- \( n(A) = 150 \) (steak), \( n(B) = 195 \) (chicken), \( n(C) = 170 \) (pork)
- \( n(A \cap B) = 99 \), \( n(B \cap C) = 103 \), \( n(A \cap C) = 62 \)
- \( n(A \cap B \cap C) = 56 \)

#### Venn Diagram Explanation:
Using the inclusion-exclusion principle and the given intersections, we can fill out the diagram, noting overlaps between the sets.

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### 3. Set Operations

#### (a) \( A \cap (B \cup C) \)
- Apply distributive laws: \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)

#### (b) \( B \cap A' \)
- Elements in \( B \) but not in \( A \).

#### (c) \( (B \cap C)' \cap A \)
- Elements not in \( B \cap C \), intersected with \( A \).

#### (d) \( C' \)
- All elements not in \( C \).

#### (e) \( (A - B) \cup C \)
- Difference of \( A \) and \( B \), then union with \( C \).

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### 4. Survey on Food Preferences (Hamburgers, Hot Dogs, Pizza)

Given:
- \( n(U) = 400 \)
- \( n(A) = 150 \) (hamburgers), \( n(B) = 140 \) (hot dogs), \( n(C) = 120 \) (pizza)
- \( n(A \cap B) = 70 \), \( n(B \cap C) = 90 \), \( n(A \cap C) = 50 \), \( n(A \cap B \cap C) = 30 \)

#### Venn Diagram and Questions:
Create a Venn diagram representing the data, ensuring to subtract intersections for accurate region sizes. Answer questions like:
- \( A \cap B \), \( A \cup C \), \( B \cup C \), etc.

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### 5. Survey on Subject Preferences (Math, Science, Social Studies)

Given:
- Total: 100 people
- Likes Math: 56, Science: 43, Social Studies: 35
- Math & Science: 18, Science & Social Studies: 10, Math & Social Studies: 12, All three: 6

#### Venn Diagram:
Use the given numbers to create a three-set Venn diagram and calculate:
- Math only: \( 56 - 18 - 12 + 6 = 32 \)
- Science only: \( 43 - 18 - 10 + 6 = 21 \)
- Social Studies only: \( 35 - 12 - 10 + 6 = 17 \)

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These are the solutions to some of the problems in your document. Would you like further details or a specific explanation for a part of this set? Let me know if you need a breakdown of any specific questions.

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### Further Questions:
1. Can you explain how to construct the Venn diagram for the food preference problem?
2. What is the method to calculate people who like only one subject in the subject preference survey?
3. How does the inclusion-exclusion principle work in set theory?
4. How to visualize the solution for \( (A \cap B) \cup C \)?
5. What are the steps to solve \( A' \cap (B \cup C) \)?

### Tip:
When solving set problems involving multiple sets, always start by filling out the intersections first before calculating the individual set components.

Explore detailed solutions to various set theory and Venn diagram problems, including set operations, constructing Venn diagrams, and applying the inclusion-exclusion principle. Suitable for high school students in Grades 9-12.

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