Answer :
Answer:
Let's analyze and solve each part based on the given sets A, B, and C:
### Given Sets:
- \( A = \{ x : x = 3n, n \in \mathbb{N}, n \leq 7 \} \)
- \( B = \{ x : x = 4n, n \in \mathbb{N}, n \leq 5 \} \)
- \( C = \{ x : x = 2n, n \in \mathbb{N}, n \leq 8 \} \)
### (i) Find \( C - A \):
To find \( C - A \), we need to find elements in set C that are not in set A.
- Elements in set C: \( C = \{ 2, 4, 6, 8, 10, 12, 14, 16 \} \)
(These are multiples of 2 up to 16 because \( n \leq 8 \))
- Elements in set A: \( A = \{ 3, 6, 9, 12, 15, 18, 21 \} \)
(These are multiples of 3 up to 21 because \( n \leq 7 \))
Now, find \( C - A \):
\[
C - A = \{ 2, 4, 8, 10, 14, 16 \}
\]
### (ii) Find \( A - (C - B) \):
First, find \( C - B \):
- Elements in set B: \( B = \{ 4, 8, 12, 16, 20 \} \)
(These are multiples of 4 up to 20 because \( n \leq 5 \))
Now, find \( C - B \):
\[
C - B = \{ 2, 6, 10, 14 \}
\]
Next, find \( A - (C - B) \):
- Elements in set A: \( A = \{ 3, 6, 9, 12, 15, 18, 21 \} \)
Remove elements of \( C - B = \{ 2, 6, 10, 14 \} \) from \( A \):
\[
A - (C - B) = \{ 3, 9, 15, 18, 21 \}
\]
### (iii) Find \( A \cap (B \cap C) \) or \( (A - B) \cap (B - A) \):
First, find \( B \cap C \):
- \( B = \{ 4, 8, 12, 16, 20 \} \)
- \( C = \{ 2, 4, 6, 8, 10, 12, 14, 16 \} \)
Intersection of B and C:
\[
B \cap C = \{ 4, 8, 12, 16 \}
\]
Now, find \( A \cap (B \cap C) \):
- \( A = \{ 3, 6, 9, 12, 15, 18, 21 \} \)
Intersection of A and \( B \cap C = \{ 4, 8, 12, 16 \} \):
\[
A \cap (B \cap C) = \{ 12 \}
\]
Therefore, the answers are:
- (i) \( C - A = \{ 2, 4, 8, 10, 14, 16 \} \)
- (ii) \( A - (C - B) = \{ 3, 9, 15, 18, 21 \} \)
- (iii) \( (A - B) \cap (B - A) = \{ 12 \} \)
Step-by-step explanation:
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