Let `A = {1, 2, 3, 4}, B = {2, 3, 5, 7} and C = {1, 3, 5, 6}`. Then, the value of `A cap ( B Delta C )` is

Options

A. `{ 1, 2, 3}`
B. `{ 1, 2}`
C. `{3, 5, 6}`
D. `{ 2, 3, 4}`

Answer

To solve the problem, we need to find the value of \( A \cap (B \Delta C) \) where \( A = \{1, 2, 3, 4\} \), \( B = \{2, 3, 5, 7\} \), and \( C = \{1, 3, 5, 6\} \).

### Step 1: Calculate \( B \Delta C \)
The symmetric difference \( B \Delta C \) is defined as:
\[ B \Delta C = (B - C) \cup (C - B) \]

#### Step 1.1: Calculate \( B - C \)
We need to find the elements in \( B \) that are not in \( C \):
\[ B = \{2, 3, 5, 7\} \]
\[ C = \{1, 3, 5, 6\} \]
The common elements between \( B \) and \( C \) are \( 3 \) and \( 5 \). Therefore, we subtract these from \( B \):
\[ B - C = \{2, 7\} \]

#### Step 1.2: Calculate \( C - B \)
Next, we find the elements in \( C \) that are not in \( B \):
\[ C = \{1, 3, 5, 6\} \]
\[ B = \{2, 3, 5, 7\} \]
The common elements between \( C \) and \( B \) are \( 3 \) and \( 5 \). Therefore, we subtract these from \( C \):
\[ C - B = \{1, 6\} \]

#### Step 1.3: Combine the results
Now we can find \( B \Delta C \):
\[ B \Delta C = (B - C) \cup (C - B) = \{2, 7\} \cup \{1, 6\} = \{1, 2, 6, 7\} \]

### Step 2: Calculate \( A \cap (B \Delta C) \)
Now we need to find the intersection of \( A \) with \( B \Delta C \):
\[ A = \{1, 2, 3, 4\} \]
\[ B \Delta C = \{1, 2, 6, 7\} \]
The common elements between \( A \) and \( B \Delta C \) are \( 1 \) and \( 2 \):
\[ A \cap (B \Delta C) = \{1, 2\} \]

### Final Answer
Thus, the value of \( A \cap (B \Delta C) \) is:
\[ \{1, 2\} \]