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

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.