The problem you provided involves sets and operations on sets. Let me break down and solve the given questions one by one.

Given:

• $S = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15\}$
• $A$ is the set of integers between -3 and 9:
  $A = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$
• $B = \{x \mid 0 \leq x \leq 8, x \in A\}$, so $B = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$
• $C = \{x \mid 1 \leq x < 15, x \in P\}$, and since $P$ is the set of primes less than 15:
  $P = \{2, 3, 5, 7, 11, 13\}$,
  thus $C = \{2, 3, 5, 7, 11, 13\}$

Now let's work through each question:

a. $A \cup (B \cap C)$

• $B \cap C$ is the intersection of sets $B$ and $C$, so $B \cap C = \{2, 3, 5, 7\}$
• $A \cup (B \cap C) = A \cup \{2, 3, 5, 7\} = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$

Thus, the result is: $A \cup (B \cap C) = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$

b. $(B \cap (A \cup C))^C$

First, find $A \cup C$:

• $A \cup C = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\}$

Next, find $B \cap (A \cup C)$:

• $B = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$
• $B \cap (A \cup C) = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$ (since $B \subseteq A \cup C$)

Now, take the complement relative to $S$:

• $B^C = S \setminus B = \{-3, -2, -1, 9, 10, 11, 12, 13, 14, 15\}$

So, $(B \cap (A \cup C))^C = \{-3, -2, -1, 9, 10, 11, 12, 13, 14, 15\}$

c. $(A - B) + C$

• $A - B = \{-3, -2, -1, 9\}$ (elements in $A$ but not in $B$)
• Now, add $C = \{2, 3, 5, 7, 11, 13\}$
• The union $(A - B) + C = \{-3, -2, -1, 9, 2, 3, 5, 7, 11, 13\}$

d. $(A \cap (B - C))^C$

• First, find $B - C = \{0, 1, 4, 6, 8\}$ (elements in $B$ but not in $C$)
• Then, $A \cap (B - C) = \{0, 1, 4, 6, 8\}$
• Now, the complement relative to $S$:
  $(A \cap (B - C))^C = S \setminus \{0, 1, 4, 6, 8\} = \{-3, -2, -1, 2, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15\}$

e. Relationship between $B$ and $C$

• $B = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}$
• $C = \{2, 3, 5, 7, 11, 13\}$
• $B \cap C = \{2, 3, 5, 7\}$, which means that $C$ is partially contained in $B$.
  Specifically, $C \subset B$ only for the elements $2, 3, 5, 7$.

Do you need any further clarification or detailed explanation for any specific part? Here are five related questions that can help you explore further:

1. What is the meaning of set complement and how is it applied?
2. How can we graphically represent the relationship between sets $A$, $B$, and $C$?
3. How do set operations relate to logical operations like AND, OR, and NOT?
4. Can you explain how to find the union and intersection of more than two sets?
5. How do Venn diagrams help in understanding set relations?

Tip: When dealing with set complements, always remember that the complement of a set is taken relative to the universal set (in this case, $S$).