If `A={x in W: 5 lt x lt 10}, B={3,4,5,6,7}` and `C={x =2n, n in N " and " n le 4}`. Find:
(i) `A cap (B cup C)`, (ii) `(B cup A) cap (B cup C)`
(iii) `B cup (A cap C)`, (iv) `(A cap B) cup (A cap C)`
Name the sets which are equal.

Answer

To solve the given problem step by step, we will first define the sets A, B, and C based on the information provided.

### Step 1: Define the sets
1. **Set A**: The set A is defined as \( A = \{ x \in W : 5 < x < 10 \} \). Since \( W \) represents whole numbers, the elements of set A are:
- \( A = \{ 6, 7, 8, 9 \} \)

2. **Set B**: The set B is given as:
- \( B = \{ 3, 4, 5, 6, 7 \} \)

3. **Set C**: The set C is defined as \( C = \{ x = 2n : n \in N \text{ and } n \leq 4 \} \). The natural numbers \( n \) can be \( 1, 2, 3, 4 \), giving us:
- For \( n = 1 \), \( x = 2 \)
- For \( n = 2 \), \( x = 4 \)
- For \( n = 3 \), \( x = 6 \)
- For \( n = 4 \), \( x = 8 \)
- Thus, \( C = \{ 2, 4, 6, 8 \} \)

### Step 2: Find the required sets
Now we will calculate the required sets one by one.

#### (i) \( A \cap (B \cup C) \)
1. **Find \( B \cup C \)**:
- \( B \cup C = \{ 3, 4, 5, 6, 7 \} \cup \{ 2, 4, 6, 8 \} \)
- Combining these, we get \( B \cup C = \{ 2, 3, 4, 5, 6, 7, 8 \} \)

2. **Find \( A \cap (B \cup C) \)**:
- \( A \cap (B \cup C) = \{ 6, 7, 8, 9 \} \cap \{ 2, 3, 4, 5, 6, 7, 8 \} \)
- The common elements are \( \{ 6, 7, 8 \} \)

#### (ii) \( (B \cup A) \cap (B \cup C) \)
1. **Find \( B \cup A \)**:
- \( B \cup A = \{ 3, 4, 5, 6, 7 \} \cup \{ 6, 7, 8, 9 \} \)
- Thus, \( B \cup A = \{ 3, 4, 5, 6, 7, 8, 9 \} \)

2. **Find \( (B \cup A) \cap (B \cup C) \)**:
- \( (B \cup A) \cap (B \cup C) = \{ 3, 4, 5, 6, 7, 8, 9 \} \cap \{ 2, 3, 4, 5, 6, 7, 8 \} \)
- The common elements are \( \{ 3, 4, 5, 6, 7, 8 \} \)

#### (iii) \( B \cup (A \cap C) \)
1. **Find \( A \cap C \)**:
- \( A \cap C = \{ 6, 7, 8, 9 \} \cap \{ 2, 4, 6, 8 \} \)
- The common elements are \( \{ 6, 8 \} \)

2. **Find \( B \cup (A \cap C) \)**:
- \( B \cup (A \cap C) = \{ 3, 4, 5, 6, 7 \} \cup \{ 6, 8 \} \)
- Thus, \( B \cup (A \cap C) = \{ 3, 4, 5, 6, 7, 8 \} \)

#### (iv) \( (A \cap B) \cup (A \cap C) \)
1. **Find \( A \cap B \)**:
- \( A \cap B = \{ 6, 7, 8, 9 \} \cap \{ 3, 4, 5, 6, 7 \} \)
- The common elements are \( \{ 6, 7 \} \)

2. **Find \( A \cap C \)** (already calculated):
- \( A \cap C = \{ 6, 8 \} \)

3. **Find \( (A \cap B) \cup (A \cap C) \)**:
- \( (A \cap B) \cup (A \cap C) = \{ 6, 7 \} \cup \{ 6, 8 \} \)
- Thus, \( (A \cap B) \cup (A \cap C) = \{ 6, 7, 8 \} \)

### Step 3: Identify equal sets
Now we will compare the results of the sets we found:
- \( A \cap (B \cup C) = \{ 6, 7, 8 \} \)
- \( (B \cup A) \cap (B \cup C) = \{ 3, 4, 5, 6, 7, 8 \} \)
- \( B \cup (A \cap C) = \{ 3, 4, 5, 6, 7, 8 \} \)
- \( (A \cap B) \cup (A \cap C) = \{ 6, 7, 8 \} \)

### Equal Sets
The equal sets are:
- \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) = \{ 6, 7, 8 \} \)