Let A = {3,6,9,12,15,18,21},
B = {4,8,12,16,20},
C = {2,4,6,8,10,12,14,16}and
D ={5,10,15,20}.
Find : (i) A-B
(ii) A-C
(iii) A-D
(iv) B-A
(v) C-A
(vi) D-A
(vii)B-C
(viii) B-D
(ix) C-B
(x) D-8
(xi) C-D
(xii) D-C.

Answer

To solve the given problem, we will find the differences between the sets A, B, C, and D as specified in the question. The operation \(X - Y\) means we need to find all elements in set \(X\) that are not in set \(Y\).

### Step-by-Step Solution:

1. **Find \(A - B\)**:
- Set A: {3, 6, 9, 12, 15, 18, 21}
- Set B: {4, 8, 12, 16, 20}
- Elements of A not in B:
- 3 (not in B)
- 6 (not in B)
- 9 (not in B)
- 12 (in B, so not included)
- 15 (not in B)
- 18 (not in B)
- 21 (not in B)
- **Result**: \(A - B = \{3, 6, 9, 15, 18, 21\}\)

2. **Find \(A - C\)**:
- Set C: {2, 4, 6, 8, 10, 12, 14, 16}
- Elements of A not in C:
- 3 (not in C)
- 6 (in C, so not included)
- 9 (not in C)
- 12 (in C, so not included)
- 15 (not in C)
- 18 (not in C)
- 21 (not in C)
- **Result**: \(A - C = \{3, 9, 15, 18, 21\}\)

3. **Find \(A - D\)**:
- Set D: {5, 10, 15, 20}
- Elements of A not in D:
- 3 (not in D)
- 6 (not in D)
- 9 (not in D)
- 12 (not in D)
- 15 (in D, so not included)
- 18 (not in D)
- 21 (not in D)
- **Result**: \(A - D = \{3, 6, 9, 12, 18, 21\}\)

4. **Find \(B - A\)**:
- Elements of B not in A:
- 4 (not in A)
- 8 (not in A)
- 12 (in A, so not included)
- 16 (not in A)
- 20 (not in A)
- **Result**: \(B - A = \{4, 8, 16, 20\}\)

5. **Find \(C - A\)**:
- Elements of C not in A:
- 2 (not in A)
- 4 (not in A)
- 6 (in A, so not included)
- 8 (not in A)
- 10 (not in A)
- 12 (in A, so not included)
- 14 (not in A)
- 16 (not in A)
- **Result**: \(C - A = \{2, 4, 8, 10, 14, 16\}\)

6. **Find \(D - A\)**:
- Elements of D not in A:
- 5 (not in A)
- 10 (not in A)
- 15 (in A, so not included)
- 20 (not in A)
- **Result**: \(D - A = \{5, 10, 20\}\)

7. **Find \(B - C\)**:
- Elements of B not in C:
- 4 (in C, so not included)
- 8 (in C, so not included)
- 12 (in C, so not included)
- 16 (in C, so not included)
- 20 (not in C)
- **Result**: \(B - C = \{20\}\)

8. **Find \(B - D\)**:
- Elements of B not in D:
- 4 (not in D)
- 8 (not in D)
- 12 (not in D)
- 16 (not in D)
- 20 (in D, so not included)
- **Result**: \(B - D = \{4, 8, 12, 16\}\)

9. **Find \(C - B\)**:
- Elements of C not in B:
- 2 (not in B)
- 4 (in B, so not included)
- 6 (not in B)
- 8 (in B, so not included)
- 10 (not in B)
- 12 (in B, so not included)
- 14 (not in B)
- 16 (in B, so not included)
- **Result**: \(C - B = \{2, 6, 10, 14\}\)

10. **Find \(D - B\)**:
- Elements of D not in B:
- 5 (not in B)
- 10 (not in B)
- 15 (not in B)
- 20 (in B, so not included)
- **Result**: \(D - B = \{5, 10, 15\}\)

11. **Find \(C - D\)**:
- Elements of C not in D:
- 2 (not in D)
- 4 (not in D)
- 6 (not in D)
- 8 (not in D)
- 10 (in D, so not included)
- 12 (not in D)
- 14 (not in D)
- 16 (in D, so not included)
- **Result**: \(C - D = \{2, 4, 6, 8, 12, 14\}\)

12. **Find \(D - C\)**:
- Elements of D not in C:
- 5 (not in C)
- 10 (in C, so not included)
- 15 (not in C)
- 20 (not in C)
- **Result**: \(D - C = \{5, 15, 20\}\)

### Summary of Results:
1. \(A - B = \{3, 6, 9, 15, 18, 21\}\)
2. \(A - C = \{3, 9, 15, 18, 21\}\)
3. \(A - D = \{3, 6, 9, 12, 18, 21\}\)
4. \(B - A = \{4, 8, 16, 20\}\)
5. \(C - A = \{2, 4, 8, 10, 14, 16\}\)
6. \(D - A = \{5, 10, 20\}\)
7. \(B - C = \{20\}\)
8. \(B - D = \{4, 8, 12, 16\}\)
9. \(C - B = \{2, 6, 10, 14\}\)
10. \(D - B = \{5, 10, 15\}\)
11. \(C - D = \{2, 4, 6, 8, 12, 14\}\)
12. \(D - C = \{5, 15, 20\}\)