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ejercicios de conjuntos y diagrmas de venn

The image you've provided contains a series of problems related to set theory, specifically focusing on operations like union (\(\cup\)), intersection (\(\cap\)), and complement (\('\)). I'll walk through the solutions for each problem.

### Let's go through each exercise step-by-step:

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**1. Given:**
- \( A = \{a, b, c\} \)
- \( B = \{c, d, e, f\} \)
- \( C = \{a, c, f\} \)
- \( D = \{g, k\} \)

**Calculate \( A \cup B \):**

- \( A \cup B = \{a, b, c\} \cup \{c, d, e, f\} \)
- This is the union of sets \( A \) and \( B \), which includes all elements from both sets, without repetition.

**Solution:** \( A \cup B = \{a, b, c, d, e, f\} \)

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**2. Given the same sets:**

**Calculate \( A \cap C \):**

- \( A \cap C = \{a, b, c\} \cap \{a, c, f\} \)
- This is the intersection of sets \( A \) and \( C \), which includes only the elements that are in both sets.

**Solution:** \( A \cap C = \{a, c\} \)

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**3. Given the same sets:**

**Calculate \( B \cup (A \cap C) \):**

- We already know \( A \cap C = \{a, c\} \).
- \( B \cup (A \cap C) = \{c, d, e, f\} \cup \{a, c\} \)

**Solution:** \( B \cup (A \cap C) = \{a, c, d, e, f\} \)

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**4. Given the same sets:**

**Calculate \( (A \cup C) \cap B \):**

- First, \( A \cup C = \{a, b, c, f\} \).
- Then, \( (A \cup C) \cap B = \{a, b, c, f\} \cap \{c, d, e, f\} \).

**Solution:** \( (A \cup C) \cap B = \{c, f\} \)

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**5. Given the same sets:**

**Calculate \( (A \cap D) \cup (B \cap C) \):**

- \( A \cap D = \{a, b, c\} \cap \{g, k\} = \emptyset \) (no common elements).
- \( B \cap C = \{c, d, e, f\} \cap \{a, c, f\} = \{c, f\} \).

**Solution:** \( (A \cap D) \cup (B \cap C) = \emptyset \cup \{c, f\} = \{c, f\} \)

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**6. In this problem:**

**Given sets:**
- \( U = \{1, 2, 3, 4, 5, 6, 7\} \)
- \( A = \{2, 4, 6\} \)
- \( B = \{1, 2, 3\} \)
- \( C = \{2, 3, 4, 7\} \)

**Calculate \( (A \cup B) \cap C \):**

- \( A \cup B = \{2, 4, 6\} \cup \{1, 2, 3\} = \{1, 2, 3, 4, 6\} \).
- Then, \( (A \cup B) \cap C = \{1, 2, 3, 4, 6\} \cap \{2, 3, 4, 7\} \).

**Solution:** \( (A \cup B) \cap C = \{2, 3, 4\} \)

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Would you like more details on any of these steps, or do you have any questions? Here are five questions related to this content:

1. What is the complement of a set and how is it calculated?
2. How does the distributive property apply to set operations like union and intersection?
3. Can you find the symmetric difference between two sets \( A \) and \( B \)?
4. What are the common properties of union and intersection operations?
5. How would you represent these operations visually using Venn diagrams?

**Tip:** When dealing with multiple set operations, it helps to break down the problem into smaller parts, solve each part, and then combine the results systematically.

Explore solutions to set theory exercises and Venn diagram problems covering operations like union, intersection, and complement. Detailed step-by-step solutions provided with examples and applications. Suitable for high school students and anyone interested in mastering set theory concepts.

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