To solve the problem step by step, we will find the required unions and intersections of the sets A, B, and C.
### Given Sets:
- \( A = \{5, 6, 7, 8, 9\} \)
- \( B = \{x : 3 < x < 8 \text{ and } x \in W\} \) (where W is the set of whole numbers)
- \( C = \{x : x \leq 5 \text{ and } x \in N\} \) (where N is the set of natural numbers)
### Step 1: Determine the elements of sets B and C
1. **Set B**: The whole numbers greater than 3 and less than 8 are \( 4, 5, 6, 7 \).
- Thus, \( B = \{4, 5, 6, 7\} \).
2. **Set C**: The natural numbers less than or equal to 5 are \( 1, 2, 3, 4, 5 \).
- Thus, \( C = \{1, 2, 3, 4, 5\} \).
### Step 2: Find the required unions and intersections
#### (i) Find \( A \cup B \) and \( (A \cup B) \cup C \)
- **Union \( A \cup B \)**:
- Combine elements of A and B:
- \( A \cup B = \{5, 6, 7, 8, 9\} \cup \{4, 5, 6, 7\} = \{4, 5, 6, 7, 8, 9\} \).
- **Union \( (A \cup B) \cup C \)**:
- Combine \( A \cup B \) with C:
- \( (A \cup B) \cup C = \{4, 5, 6, 7, 8, 9\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).
#### (ii) Find \( B \cup C \) and \( A \cup (B \cup C) \)
- **Union \( B \cup C \)**:
- Combine elements of B and C:
- \( B \cup C = \{4, 5, 6, 7\} \cup \{1, 2, 3, 4, 5\} = \{1, 2, 3, 4, 5, 6, 7\} \).
- **Union \( A \cup (B \cup C) \)**:
- Combine A with \( B \cup C \):
- \( A \cup (B \cup C) = \{5, 6, 7, 8, 9\} \cup \{1, 2, 3, 4, 5, 6, 7\} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).
#### (iii) Find \( A \cap B \) and \( (A \cap B) \cap C \)
- **Intersection \( A \cap B \)**:
- Find common elements in A and B:
- \( A \cap B = \{5, 6, 7, 8, 9\} \cap \{4, 5, 6, 7\} = \{5, 6, 7\} \).
- **Intersection \( (A \cap B) \cap C \)**:
- Find common elements in \( A \cap B \) and C:
- \( (A \cap B) \cap C = \{5, 6, 7\} \cap \{1, 2, 3, 4, 5\} = \{5\} \).
#### (iv) Find \( B \cap C \) and \( A \cap (B \cap C) \)
- **Intersection \( B \cap C \)**:
- Find common elements in B and C:
- \( B \cap C = \{4, 5, 6, 7\} \cap \{1, 2, 3, 4, 5\} = \{4, 5\} \).
- **Intersection \( A \cap (B \cap C) \)**:
- Find common elements in A and \( B \cap C \):
- \( A \cap (B \cap C) = \{5, 6, 7, 8, 9\} \cap \{4, 5\} = \{5\} \).
### Step 3: Check the equality of unions and intersections
- **Check if \( (A \cup B) \cup C = A \cup (B \cup C) \)**:
- Both equal to \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \), so they are equal.
- **Check if \( (A \cap B) \cap C = A \cap (B \cap C) \)**:
- Both equal to \( \{5\} \), so they are equal.
### Final Results:
1. \( A \cup B = \{4, 5, 6, 7, 8, 9\} \)
2. \( (A \cup B) \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
3. \( B \cup C = \{1, 2, 3, 4, 5, 6, 7\} \)
4. \( A \cup (B \cup C) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
5. \( A \cap B = \{5, 6, 7\} \)
6. \( (A \cap B) \cap C = \{5\} \)
7. \( B \cap C = \{4, 5\} \)
8. \( A \cap (B \cap C) = \{5\} \)
