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Let's go through the given sets and determine the required operations.

The universal set \( U \) is:
\[ U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \]

The given sets are:
- \( A = \{1, 9, 13, 17\} \)
- \( B = \{7, 11, 17\} \)
- \( C = \{3, 5, 15\} \)

We will now solve the following:

### a) \( A \cap B \)
The intersection \( A \cap B \) is the set of elements that are in both \( A \) and \( B \).

- \( A = \{1, 9, 13, 17\} \)
- \( B = \{7, 11, 17\} \)

Common elements: \( \{17\} \)

Thus, \( A \cap B = \{17\} \).

### b) \( A \cap C \)
The intersection \( A \cap C \) is the set of elements common to both \( A \) and \( C \).

- \( A = \{1, 9, 13, 17\} \)
- \( C = \{3, 5, 15\} \)

No common elements.

Thus, \( A \cap C = \emptyset \) (empty set).

### c) \( B \cap U \)
The intersection \( B \cap U \) is the set of elements common to \( B \) and the universal set \( U \). Since \( B \) is already a subset of \( U \), we know:

\( B \cap U = B = \{7, 11, 17\} \).

### d) \( (C \cap A)' \)
The complement of \( C \cap A \), which is \( (C \cap A)' \), means we first find \( C \cap A \), and then we take its complement with respect to the universal set \( U \).

From part b), we found \( C \cap A = \emptyset \).

The complement of \( \emptyset \) in \( U \) is the entire universal set \( U \):

So, \( (C \cap A)' = U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \).

### e) \( A \cup U \)
The union \( A \cup U \) is the set of elements in either \( A \) or \( U \) (or both). Since \( A \) is already a subset of \( U \), the result is:

\( A \cup U = U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \).

### f) \( A \cap (B \cap C') \)
First, we need to find \( C' \) (the complement of \( C \) in \( U \)):

\( C = \{3, 5, 15\} \), so \( C' = U - C = \{1, 7, 9, 11, 13, 17\} \).

Now, find \( B \cap C' \):

- \( B = \{7, 11, 17\} \)
- \( C' = \{1, 7, 9, 11, 13, 17\} \)

Common elements: \( B \cap C' = \{7, 11, 17\} \).

Finally, find \( A \cap (B \cap C') \):

- \( A = \{1, 9, 13, 17\} \)
- \( B \cap C' = \{7, 11, 17\} \)

Common element: \( \{17\} \).

So, \( A \cap (B \cap C') = \{17\} \).

### g) \( n(A \cap B') \)
First, find \( B' \) (the complement of \( B \) in \( U \)):

\( B = \{7, 11, 17\} \), so \( B' = U - B = \{1, 3, 5, 9, 13, 15\} \).

Now, find \( A \cap B' \):

- \( A = \{1, 9, 13, 17\} \)
- \( B' = \{1, 3, 5, 9, 13, 15\} \)

Common elements: \( A \cap B' = \{1, 9, 13\} \).

The number of elements in \( A \cap B' \) is 3.

Thus, \( n(A \cap B') = 3 \).

### h) \( n(A \cup B \cup C) \)
The union \( A \cup B \cup C \) is the set of elements in \( A \), \( B \), or \( C \) (or all). Let's combine all the sets:

- \( A = \{1, 9, 13, 17\} \)
- \( B = \{7, 11, 17\} \)
- \( C = \{3, 5, 15\} \)

Union: \( A \cup B \cup C = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} \).

The number of elements in this union is 9.

Thus, \( n(A \cup B \cup C) = 9 \).

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Would you like further clarification on any part of this? 

Here are five related questions to deepen your understanding:
1. How do complements and intersections differ in set operations?
2. What happens when we take the union of a set with the universal set?
3. Can an empty set have a complement?
4. Why is \( A \cap B' \) not the same as \( A \cap B \)'?
5. How would the results change if the universal set were different?

**Tip:** Remember, the complement of a set refers to everything in the universal set except the elements of the given set.

Given that U = {1, 3, 5, 7, 9, 11, 13, 15, 17}, A = {1, 9, 13, 17}, B = {7, 11, 17}, and C = {3, 5, 15}, determine the following: a) A ∩ B, b) A ∩ C, c) B ∩ U, d) (C ∩ A)', e) A ∪ U, f) A ∩ (B ∩ C'), g) n(A ∩ B'), h) n(A ∪ B ∪ C).

Solve set theory problems involving the universal set U = {1, 3, 5, 7, 9, 11, 13, 15, 17} and sets A, B, and C. This includes intersections, unions, complements, and calculating the number of elements in different combinations. Ideal for high school students (Grades 9-12).

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