If n (A) = 115. n(B) = 326, n(A - B) = 47, then what is `n( A cup B )` equal to?

Options

A. 373
B. 165
C. 370
D. 394

Answer

To solve the problem, we need to find the value of \( n(A \cup B) \) using the given values.

### Step-by-step Solution:

1. **Understand the Given Values:**
- \( n(A) = 115 \)
- \( n(B) = 326 \)
- \( n(A - B) = 47 \)

2. **Use the Formula for Union of Two Sets:**
The formula for the union of two sets is given by:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Here, \( n(A \cap B) \) is the number of elements in both sets A and B.

3. **Find \( n(A \cap B) \):**
We know that:
\[ n(A) = n(A - B) + n(A \cap B) \]
Rearranging this gives us:
\[ n(A \cap B) = n(A) - n(A - B) \]
Substituting the values:
\[ n(A \cap B) = 115 - 47 = 68 \]

4. **Substitute Values into the Union Formula:**
Now we can substitute \( n(A) \), \( n(B) \), and \( n(A \cap B) \) into the union formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
\[ n(A \cup B) = 115 + 326 - 68 \]

5. **Calculate the Result:**
Performing the calculation:
\[ n(A \cup B) = 441 - 68 = 373 \]

6. **Final Answer:**
Therefore, the value of \( n(A \cup B) \) is \( 373 \).