Let's go through each part of the problem using the given sets.

Given:

• $U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$
• $A = \{1, 9, 13, 17\}$
• $B = \{1, 7, 11, 17\}$
• $C = \{3, 5, 15\}$

Now let's calculate each of the following:

a) $A \cap B$ (Intersection of A and B)

This is the set of elements common to both $A$ and $B$. $A \cap B = \{1, 9, 13, 17\} \cap \{1, 7, 11, 17\} = \{1, 17\}$ Answer: $\{1, 17\}$

b) $A \cap C$ (Intersection of A and C)

This is the set of elements common to both $A$ and $C$. $A \cap C = \{1, 9, 13, 17\} \cap \{3, 5, 15\} = \emptyset$ Answer: $\emptyset$ (the empty set)

c) $B \cap U$ (Intersection of B and U)

This is the set of elements common to both $B$ and the universal set $U$. Since $B$ is a subset of $U$, their intersection is $B$ itself. $B \cap U = B = \{1, 7, 11, 17\}$ Answer: $\{1, 7, 11, 17\}$

d) $(C \cap A)'$ (Complement of $C \cap A$ relative to U)

First, calculate $C \cap A$: $C \cap A = \{3, 5, 15\} \cap \{1, 9, 13, 17\} = \emptyset$ The complement of the empty set relative to $U$ is $U$ itself: $(C \cap A)' = U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$ Answer: $\{1, 3, 5, 7, 9, 11, 13, 15, 17\}$

e) $A \cup U$ (Union of A and U)

Since $U$ is the universal set, the union of $A$ and $U$ is $U$ itself: $A \cup U = U = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$ Answer: $\{1, 3, 5, 7, 9, 11, 13, 15, 17\}$

f) $A \cap (B \cap C')$

First, find $C'$ (complement of $C$): $C' = U - C = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} - \{3, 5, 15\} = \{1, 7, 9, 11, 13, 17\}$ Next, calculate $B \cap C'$: $B \cap C' = \{1, 7, 11, 17\} \cap \{1, 7, 9, 11, 13, 17\} = \{1, 7, 11, 17\}$ Now, calculate $A \cap (B \cap C')$: $A \cap \{1, 7, 11, 17\} = \{1, 9, 13, 17\} \cap \{1, 7, 11, 17\} = \{1, 17\}$ Answer: $\{1, 17\}$

g) $n(A \cap B')$ (Cardinality of $A \cap B'$)

First, find $B'$ (complement of $B$): $B' = U - B = \{1, 3, 5, 7, 9, 11, 13, 15, 17\} - \{1, 7, 11, 17\} = \{3, 5, 9, 13, 15\}$ Now, calculate $A \cap B'$: $A \cap B' = \{1, 9, 13, 17\} \cap \{3, 5, 9, 13, 15\} = \{9, 13\}$ The cardinality $n(A \cap B')$ is the number of elements in $\{9, 13\}$, which is 2. Answer: 2

h) $n(A \cup B \cup C)$ (Cardinality of $A \cup B \cup C$)

First, calculate $A \cup B \cup C$: $A \cup B = \{1, 9, 13, 17\} \cup \{1, 7, 11, 17\} = \{1, 7, 9, 11, 13, 17\}$ $A \cup B \cup C = \{1, 7, 9, 11, 13, 17\} \cup \{3, 5, 15\} = \{1, 3, 5, 7, 9, 11, 13, 15, 17\}$ The cardinality $n(A \cup B \cup C)$ is the number of elements in $\{1, 3, 5, 7, 9, 11, 13, 15, 17\}$, which is 9. Answer: 9

Do you have any questions or need further details on any step?

Here are 5 related questions to explore further:

1. What is the difference between the union and intersection of sets?
2. How is the complement of a set calculated with respect to the universal set?
3. How does the cardinality of sets help in determining the size of combined sets?
4. What happens when you take the complement of the intersection of two sets?
5. How would you represent the symmetric difference between sets A and B?

Tip: The complement of a set contains all elements in the universal set that are not in the original set.