To solve the problem, we will perform set subtraction for the given sets \( A, B, C, \) and \( D \). The set subtraction \( X - Y \) means we will find all elements in set \( X \) that are not in set \( Y \).
Given sets:
- \( A = \{3, 6, 12, 15, 18, 21\} \)
- \( B = \{4, 8, 12, 16, 20\} \)
- \( C = \{2, 4, 6, 10, 12, 14, 16\} \)
- \( D = \{5, 10, 15, 20\} \)
We will compute the following:
1. \( A - B \)
2. \( B - C \)
3. \( C - D \)
4. \( D - C \)
### Step-by-Step Solution:
#### (i) Calculate \( A - B \):
- Elements of \( A \): \( 3, 6, 12, 15, 18, 21 \)
- Elements of \( B \): \( 4, 8, 12, 16, 20 \)
Now, we check each element of \( A \):
- \( 3 \) is not in \( B \) → include \( 3 \)
- \( 6 \) is not in \( B \) → include \( 6 \)
- \( 12 \) is in \( B \) → do not include \( 12 \)
- \( 15 \) is not in \( B \) → include \( 15 \)
- \( 18 \) is not in \( B \) → include \( 18 \)
- \( 21 \) is not in \( B \) → include \( 21 \)
Thus, \( A - B = \{3, 6, 15, 18, 21\} \).
#### (ii) Calculate \( B - C \):
- Elements of \( B \): \( 4, 8, 12, 16, 20 \)
- Elements of \( C \): \( 2, 4, 6, 10, 12, 14, 16 \)
Now, we check each element of \( B \):
- \( 4 \) is in \( C \) → do not include \( 4 \)
- \( 8 \) is not in \( C \) → include \( 8 \)
- \( 12 \) is in \( C \) → do not include \( 12 \)
- \( 16 \) is in \( C \) → do not include \( 16 \)
- \( 20 \) is not in \( C \) → include \( 20 \)
Thus, \( B - C = \{8, 20\} \).
#### (iii) Calculate \( C - D \):
- Elements of \( C \): \( 2, 4, 6, 10, 12, 14, 16 \)
- Elements of \( D \): \( 5, 10, 15, 20 \)
Now, we check each element of \( C \):
- \( 2 \) is not in \( D \) → include \( 2 \)
- \( 4 \) is not in \( D \) → include \( 4 \)
- \( 6 \) is not in \( D \) → include \( 6 \)
- \( 10 \) is in \( D \) → do not include \( 10 \)
- \( 12 \) is not in \( D \) → include \( 12 \)
- \( 14 \) is not in \( D \) → include \( 14 \)
- \( 16 \) is not in \( D \) → include \( 16 \)
Thus, \( C - D = \{2, 4, 6, 12, 14, 16\} \).
#### (iv) Calculate \( D - C \):
- Elements of \( D \): \( 5, 10, 15, 20 \)
- Elements of \( C \): \( 2, 4, 6, 10, 12, 14, 16 \)
Now, we check each element of \( D \):
- \( 5 \) is not in \( C \) → include \( 5 \)
- \( 10 \) is in \( C \) → do not include \( 10 \)
- \( 15 \) is not in \( C \) → include \( 15 \)
- \( 20 \) is not in \( C \) → include \( 20 \)
Thus, \( D - C = \{5, 15, 20\} \).
### Final Results:
1. \( A - B = \{3, 6, 15, 18, 21\} \)
2. \( B - C = \{8, 20\} \)
3. \( C - D = \{2, 4, 6, 12, 14, 16\} \)
4. \( D - C = \{5, 15, 20\} \)
