If X={1,2,3,4} , Y={2,3,5,7}, Z={3,6,8,9},W={2,4,8,10}, then

Options

A. `(X Delta Y) Delta (Z Delta W)" is" {1,2,3,5,6,7,9,10}`
B. `(X Delta Z) Delta (Y Delta W)" is" {1,2,3,5,6,7,10}`
C. `(X Delta Y) Delta (Z Delta W)" is" {1,3,10}`
D. `(X Delta Y) Delta (Z Delta W)" is" {1,2,3,4,5,6,7,8,9,10}`

Answer

To solve the problem, we need to find the symmetric difference of the sets X, Y, Z, and W as described. Let's break down the solution step by step.

### Step 1: Define the Sets
We have the following sets:
- \( X = \{1, 2, 3, 4\} \)
- \( Y = \{2, 3, 5, 7\} \)
- \( Z = \{3, 6, 8, 9\} \)
- \( W = \{2, 4, 8, 10\} \)

### Step 2: Calculate \( A = X \Delta Y \)
The symmetric difference \( A \) is defined as:
\[ A = (X - Y) \cup (Y - X) \]

**Calculate \( X - Y \):**
- Elements in \( X \) but not in \( Y \):
- From \( X \): 1, 2, 3, 4
- From \( Y \): 2, 3
- Thus, \( X - Y = \{1, 4\} \)

**Calculate \( Y - X \):**
- Elements in \( Y \) but not in \( X \):
- From \( Y \): 2, 3, 5, 7
- From \( X \): 2, 3
- Thus, \( Y - X = \{5, 7\} \)

**Combine the results:**
\[ A = (X - Y) \cup (Y - X) = \{1, 4\} \cup \{5, 7\} = \{1, 4, 5, 7\} \]

### Step 3: Calculate \( B = Z \Delta W \)
Now we calculate \( B \):
\[ B = (Z - W) \cup (W - Z) \]

**Calculate \( Z - W \):**
- Elements in \( Z \) but not in \( W \):
- From \( Z \): 3, 6, 8, 9
- From \( W \): 2, 4, 8, 10
- Thus, \( Z - W = \{3, 6, 9\} \)

**Calculate \( W - Z \):**
- Elements in \( W \) but not in \( Z \):
- From \( W \): 2, 4, 8, 10
- From \( Z \): 3, 6, 8
- Thus, \( W - Z = \{2, 4, 10\} \)

**Combine the results:**
\[ B = (Z - W) \cup (W - Z) = \{3, 6, 9\} \cup \{2, 4, 10\} = \{2, 3, 4, 6, 9, 10\} \]

### Step 4: Calculate \( A \Delta B \)
Now we find the symmetric difference \( A \Delta B \):
\[ A \Delta B = (A - B) \cup (B - A) \]

**Calculate \( A - B \):**
- Elements in \( A \) but not in \( B \):
- From \( A \): 1, 4, 5, 7
- From \( B \): 2, 3, 4, 6, 9, 10
- Thus, \( A - B = \{1, 5, 7\} \)

**Calculate \( B - A \):**
- Elements in \( B \) but not in \( A \):
- From \( B \): 2, 3, 4, 6, 9, 10
- From \( A \): 1, 4, 5, 7
- Thus, \( B - A = \{2, 3, 6, 9, 10\} \)

**Combine the results:**
\[ A \Delta B = (A - B) \cup (B - A) = \{1, 5, 7\} \cup \{2, 3, 6, 9, 10\} = \{1, 2, 3, 5, 6, 7, 9, 10\} \]

### Final Result
The final result of \( A \Delta B \) is:
\[ \{1, 2, 3, 5, 6, 7, 9, 10\} \]