Let A = {1, 3, 5, 7, 8}, B = {4, 5, 7}, C = {4, 6, 7}. Explicitly write down the
sets    B \ (A ∪ C), B \ (A ∩ C), A × B.

Let's work through the three set operations step by step:

### 1. \( B \setminus (A \cup C) \)
This represents the set of elements in \( B \) that are **not** in \( A \cup C \).

- \( A \cup C \): This is the union of sets \( A \) and \( C \), which means all elements that are in \( A \), \( C \), or both.
  \[
  A \cup C = \{1, 3, 4, 5, 6, 7, 8\}
  \]
- Now, \( B \setminus (A \cup C) \): Remove all elements in \( A \cup C \) from \( B \). The elements of \( B \) are \( \{4, 5, 7\} \).
  - Since all elements of \( B \) (4, 5, 7) are in \( A \cup C \), the result is:
  \[
  B \setminus (A \cup C) = \emptyset
  \]

### 2. \( B \setminus (A \cap C) \)
This represents the set of elements in \( B \) that are **not** in \( A \cap C \).

- \( A \cap C \): This is the intersection of sets \( A \) and \( C \), meaning all elements that are common to both sets.
  \[
  A \cap C = \{7\}
  \]
- Now, \( B \setminus (A \cap C) \): Remove all elements in \( A \cap C \) from \( B \). The elements of \( B \) are \( \{4, 5, 7\} \).
  - Removing 7 from \( B \), we get:
  \[
  B \setminus (A \cap C) = \{4, 5\}
  \]

### 3. \( A \times B \)
This is the **Cartesian product** of sets \( A \) and \( B \). The Cartesian product consists of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \).

- \( A = \{1, 3, 5, 7, 8\} \)
- \( B = \{4, 5, 7\} \)

So, the Cartesian product \( A \times B \) is:
\[
A \times B = \{(1, 4), (1, 5), (1, 7), (3, 4), (3, 5), (3, 7), (5, 4), (5, 5), (5, 7), (7, 4), (7, 5), (7, 7), (8, 4), (8, 5), (8, 7)\}
\]

### Final Answer:
1. \( B \setminus (A \cup C) = \emptyset \)
2. \( B \setminus (A \cap C) = \{4, 5\} \)
3. \( A \times B = \{(1, 4), (1, 5), (1, 7), (3, 4), (3, 5), (3, 7), (5, 4), (5, 5), (5, 7), (7, 4), (7, 5), (7, 7), (8, 4), (8, 5), (8, 7)\} \)

Would you like further details on any step, or have additional questions?

### Related Questions:
1. What is the Cartesian product of two sets of different sizes?
2. How do you compute the symmetric difference between two sets?
3. What are the properties of set intersection and union?
4. Can you explain the difference between a subset and a proper subset?
5. What are the applications of Cartesian products in real-world problems?

### Tip:
When working with sets, drawing Venn diagrams can often help visualize the relationships between unions, intersections, and differences.

Explore set operations B ∖ (A ∪ C), B ∖ (A ∩ C), and Cartesian product A × B step-by-step. Understand how to find elements in sets B that are not in the union or intersection of sets A and C. Suitable for advanced high school students and those interested in set theory concepts.

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