Given:

• $U = \{0, 1, 2, \dots, 9\}$ (Universal set)
• $A = \{1, 2, 3, 4, 5\}$
• $B = \{0, 1, 2, 6, 8\}$
• $C = \{3, 6, 9\}$

We need to calculate:

1. Complement of $A$, denoted $A'$: $A' = U - A = \{0, 6, 7, 8, 9\}$

2. Difference $B / A$: $B / A = B - A = \{0, 6, 8\}$

3. Intersection of $A'$ and $C'$:

   • First, find $C'$: $C' = U - C = \{0, 1, 2, 4, 5, 7, 8\}$
   • Then find $A' \cap C'$: $A' \cap C' = \{0, 6, 7, 8, 9\} \cap \{0, 1, 2, 4, 5, 7, 8\} = \{0, 7, 8\}$

4. Cartesian product $C \times (A \cap B)$:

   • First, find $A \cap B$: $A \cap B = \{1, 2\}$
   • Then $C \times (A \cap B)$ is: $C \times (A \cap B) = \{(3, 1), (3, 2), (6, 1), (6, 2), (9, 1), (9, 2)\}$

5. Cartesian product $(A / B) \times C$:

   • First, find $A / B$: $A / B = A - B = \{3, 4, 5\}$
   • Then $(A / B) \times C$ is: $(A / B) \times C = \{(3, 3), (3, 6), (3, 9), (4, 3), (4, 6), (4, 9), (5, 3), (5, 6), (5, 9)\}$

Final Answers:

1. $A' = \{0, 6, 7, 8, 9\}$
2. $B / A = \{0, 6, 8\}$
3. $A' \cap C' = \{0, 7, 8\}$
4. $C \times (A \cap B) = \{(3, 1), (3, 2), (6, 1), (6, 2), (9, 1), (9, 2)\}$
5. $(A / B) \times C = \{(3, 3), (3, 6), (3, 9), (4, 3), (4, 6), (4, 9), (5, 3), (5, 6), (5, 9)\}$

Would you like any further explanations on these steps?

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Here are five related questions:

1. What would be $C'$ in another universal set $U' = \{0, 1, 2, ..., 12\}$?
2. Can you find $A \cap B'$ and $B \cap A'$?
3. How would $(B \cup C) \cap A'$ be evaluated?
4. Can you list the Cartesian product $B \times C$?
5. How would $A \cup C$ differ from $A \cap C$?

Tip: In set theory, the complement of a set is calculated with respect to a universal set $U$, so always confirm what $U$ is when working with complements.