https://file.aiquickdraw.com/mathbot/file/1727453233643qwwnywwx.jpg

Let's analyze and solve the given problem step-by-step:

### Provided Information
1. **Universal Set, \( U \)**: \(\{1, 2, 3, 4, 5, 6\}\)
2. **Set \( A \)**: \(\{1, 2, 3, 4\}\)
3. **Set \( B \)**: \(\{3, 5, 6\}\)
4. **Set \( C \)**: \(\{3, 4, 5, 6\}\)

The questions to solve are:
1. \((i)\) Find \((A \cup B)'\).
2. \((ii)\) Find \( A' \cap B' \).
3. \((iii)\) What do you notice?
4. \((iv)\) Prove that \((A \cap B)' = A' \cup B'\).
5. \((v)\) Find \( A \Delta B \).

### Solutions
1. **(i) \((A \cup B)'\)**:
   - \( A \cup B \) represents the union of sets \( A \) and \( B \), which includes all elements in either \( A \) or \( B \).
   \[
   A \cup B = \{1, 2, 3, 4, 5, 6\}
   \]
   - The complement, \((A \cup B)'\), includes all elements in \( U \) but not in \( A \cup B \).
   \[
   (A \cup B)' = U - (A \cup B) = \{\}
   \]
   
2. **(ii) \( A' \cap B' \)**:
   - \( A' \) is the complement of \( A \), meaning all elements in \( U \) but not in \( A \).
   \[
   A' = \{5, 6\}
   \]
   - \( B' \) is the complement of \( B \), meaning all elements in \( U \) but not in \( B \).
   \[
   B' = \{1, 2, 4\}
   \]
   - The intersection, \( A' \cap B' \), includes all elements that are in both \( A' \) and \( B' \).
   \[
   A' \cap B' = \{\}
   \]

3. **(iii) What do you notice?**
   - Notice that \( (A \cup B)' = A' \cap B' \). This is a standard property in set theory, known as De Morgan's Law.

4. **(iv) Prove that \((A \cap B)' = A' \cup B'\)**:
   - By De Morgan's Law, the complement of the intersection of two sets is equal to the union of their complements:
   \[
   (A \cap B)' = A' \cup B'
   \]
   - Let's verify:
     - \( A \cap B = \{3\} \)
     - \( A' = \{5, 6\} \) and \( B' = \{1, 2, 4\} \)
     - \( A' \cup B' = \{1, 2, 4, 5, 6\} \)
     - \( (A \cap B)' = U - \{3\} = \{1, 2, 4, 5, 6\} \)
     - This shows that \((A \cap B)' = A' \cup B'\).

5. **(v) Find \( A \Delta B \)**:
   - The symmetric difference \( A \Delta B \) includes elements in \( A \cup B \) but not in \( A \cap B \).
   \[
   A \Delta B = (A \cup B) - (A \cap B)
   \]
   - \( A \cup B = \{1, 2, 3, 4, 5, 6\} \)
   - \( A \cap B = \{3\} \)
   - Therefore:
   \[
   A \Delta B = \{1, 2, 4, 5, 6\}
   \]

Would you like me to expand further or provide more details? Here are some related questions for deeper understanding:

1. What is the relation between union and intersection of sets?
2. How can we represent De Morgan’s laws using Venn diagrams?
3. Can you prove that \((A \Delta B)' = A' \cap B'\)?
4. What are the applications of symmetric difference in probability?
5. How does the complement operation affect the cardinality of a set?

**Tip**: Always try using Venn diagrams to visualize set operations like union, intersection, and complements for a clearer understanding.

If U = {1,2,3,4,5,6}, A = {1,2,3,4}, B = {3,5,6}, C = {3,4,5,6}, find out (i) (A∪B)' (ii) A'∩B' (iii) What do you notice (iv) Prove that (A∩B)' = A'∪B' (v) Find AΔB.

Solve set theory problems involving union, intersection, complements, and symmetric difference. Explore De Morgan's Law to prove (A ∩ B)' = A' ∪ B'. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation