If X and Y are two sets such that n(X)=17,n(Y)=23 and `n(X uu Y)`=38, then `n( X nn Y)`

Options

A. 2
B. 1
C. 3
D. 4

Answer

To solve the problem, we will use the formula that relates the number of elements in the union and intersection of two sets. The formula is:

\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

Where:
- \( n(X \cup Y) \) is the number of elements in the union of sets X and Y.
- \( n(X) \) is the number of elements in set X.
- \( n(Y) \) is the number of elements in set Y.
- \( n(X \cap Y) \) is the number of elements in the intersection of sets X and Y.

### Step-by-Step Solution:

1. **Identify the given values:**
- \( n(X) = 17 \)
- \( n(Y) = 23 \)
- \( n(X \cup Y) = 38 \)

2. **Substitute the values into the formula:**
\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]
Substituting the known values:
\[ 38 = 17 + 23 - n(X \cap Y) \]

3. **Simplify the equation:**
- First, calculate \( 17 + 23 \):
\[ 17 + 23 = 40 \]
- Now substitute back into the equation:
\[ 38 = 40 - n(X \cap Y) \]

4. **Rearrange to find \( n(X \cap Y) \):**
\[ n(X \cap Y) = 40 - 38 \]
\[ n(X \cap Y) = 2 \]

5. **Conclusion:**
The number of elements in the intersection of sets X and Y is \( n(X \cap Y) = 2 \).

### Final Answer:
\( n(X \cap Y) = 2 \)