https://file.aiquickdraw.com/mathbot/file/1726699235390l8alc76m.jpg

The image contains a problem set involving sets and Venn diagrams. Here's the breakdown of the problem:

### Given sets:
- \( U = \{1, 2, 3, 4, \dots, 25\} \)
- \( A = \{2, 4, 6, 8, 10, 12, 14, 16\} \)
- \( B = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \)
- \( C = \{1, 3, 5, 7, 11, 17\} \)

You are asked to:
1. Draw a Venn diagram of these sets.
2. Determine the following operations:
   - (a) \( U \cap A \) 
   - (b) \( B \cap C \)
   - (c) \( A \cup C \)
   - (d) \( B - C \)
   - (e) \( U - A \)
   - (f) \( \overline{A} \cap \overline{B} \) (complement of A and B)
   - (g) \( B \oplus C \) (symmetric difference of B and C)
   - (h) The cardinalities \( |A|, |B|, |C| \)
   - (i) The probabilities \( P(A), P(C) \) (assuming probability is based on set size divided by universal set size).

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### Let's solve each part:

1. **(a) \( U \cap A \)**:
   - The universal set \( U \) contains all elements from 1 to 25, and \( A \) is a subset of \( U \).
   - So, \( U \cap A = A = \{2, 4, 6, 8, 10, 12, 14, 16\} \).

2. **(b) \( B \cap C \)**:
   - Intersection of \( B \) and \( C \) are the elements common to both sets.
   - \( B \cap C = \{1, 3, 5, 7, 11, 17\} \) (since \( C \) is a subset of \( B \)).

3. **(c) \( A \cup C \)**:
   - The union includes all elements from \( A \) and \( C \) without repetition.
   - \( A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 16, 17\} \).

4. **(d) \( B - C \)**:
   - This is the set of elements in \( B \) that are not in \( C \).
   - \( B - C = \{9, 13, 15\} \).

5. **(e) \( U - A \)**:
   - Elements in \( U \) but not in \( A \).
   - \( U - A = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25\} \).

6. **(f) \( \overline{A} \cap \overline{B} \)**:
   - Complement of \( A \) in \( U \) is \( \overline{A} = U - A \), and complement of \( B \) in \( U \) is \( \overline{B} = U - B \).
   - \( \overline{A} = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25\} \)
   - \( \overline{B} = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25\} \)
   - \( \overline{A} \cap \overline{B} = \{18, 19, 20, 21, 22, 23, 24, 25\} \).

7. **(g) \( B \oplus C \)**:
   - Symmetric difference includes elements in either \( B \) or \( C \), but not in both.
   - \( B \oplus C = B \cup C - (B \cap C) = \{9, 13, 15\} \).

8. **(h) \( |A|, |B|, |C| \)**:
   - The cardinality (number of elements) in each set:
     - \( |A| = 8 \) (since \( A = \{2, 4, 6, 8, 10, 12, 14, 16\} \)),
     - \( |B| = 9 \) (since \( B = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \)),
     - \( |C| = 6 \) (since \( C = \{1, 3, 5, 7, 11, 17\} \)).

9. **(i) \( P(A), P(C) \)**:
   - Assuming the probability is based on the ratio of elements to the total size of \( U \) (which contains 25 elements),
     - \( P(A) = \frac{|A|}{|U|} = \frac{8}{25} \),
     - \( P(C) = \frac{|C|}{|U|} = \frac{6}{25} \).

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Would you like more details or a specific explanation on any part?

### Related Questions:
1. What is the union \( B \cup C \)?
2. How do you interpret \( A \oplus B \) (symmetric difference between \( A \) and \( B \))?
3. What would be the result of \( U \cap B \)?
4. Can we find the complement of \( C \) in \( U \)?
5. What happens when you take the difference \( C - B \)?

### Tip:
When working with set operations, always consider if you're asked for union, intersection, or difference—each gives different types of information about the relationship between sets.

Terdapat himpunan U = {1, 2, 3, 4, ..., 25}, A = {2, 4, 6, 8, 10, 12, 14, 16}, B = {1, 3, 5, 7, 9, 11, 13, 15, 17}, C = {1, 3, 5, 7, 11, 17}. Tentukan operasi himpunan seperti U ∩ A, B ∩ C, A ∪ C, dan lainnya, serta hitung probabilitas P(A) dan P(C).

Learn how to solve set operations involving the universal set U = {1, 2, 3, ..., 25} and subsets A, B, and C. Operations include union, intersection, and set differences, along with calculating probabilities P(A) and P(C). Suitable for high school students in Grades 9-12.

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