https://file.aiquickdraw.com/mathbot/file/1726043387939kfehfyqx.jpg

U= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}
A ={0, 3, 6, 9, 12}
B = {2, 4, 6, 8, 10, 12, 14}
C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}  answer the odd numbers only 

Here are the results for the odd-numbered questions:

- **31. (A ∩ C)' = {0, 1, 2, 4, 5, 7, 8, 10, 11, 12, 13, 14}**
- **33. A' ∩ (B ∪ C)' = {11, 13}**
- **35. C' ∩ (A ∪ B)' = {11, 13}**
- **37. (A ∩ B)' ∪ (B ∩ C)' = {0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14}**
- **39. (A' ∪ B)' ∩ (U ∪ C') = {0, 3, 9}**

Let me know if you want more details about any of these problems.

Here are some related questions to expand your understanding:

1. How do we compute set complements and why are they important?
2. Can you explain how the intersection and union of sets differ in logical terms?
3. How would the results change if we altered the universal set \( U \)?
4. What is the role of De Morgan's laws in set theory?
5. Can you show how set operations can be applied in real-world problem solving?

**Tip:** Always keep track of the universal set \( U \), as it is key to calculating complements correctly.

U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}; A = {0, 3, 6, 9, 12}; B = {2, 4, 6, 8, 10, 12, 14}; C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Answer the odd-numbered questions.

Explore key set theory operations with a focus on complements, intersections, and unions. Solve problems like (A ∩ C)', (A ∪ B ∪ C)', and more using De Morgan's laws and basic set theory. Suitable for students in Grades 9-12.

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