Let U=(0,1,2,.......,9),A=(1,2,3,4,5),B=(0,1,2,6,8) and C=(3,6,9). Then find A',B/A,A' intersection c',c×(A intersection B) and (A/B)×c

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.

Let U=(0,1,2,.......,9), A=(1,2,3,4,5), B=(0,1,2,6,8) and C=(3,6,9). Then find A', B/A, A' intersection C', C×(A intersection B), and (A/B)×C.

Explore set theory with examples using universal sets, complements, differences, intersections, and Cartesian products. This problem involves calculating A', B/A, A' ∩ C', C × (A ∩ B), and (A/B) × C using the universal set U=(0,1,2,...,9) and subsets A, B, and C. Ideal for high school students in Grades 10-12.

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