If n(A) denotes the number of elements in set A and if `n(A) = 4, n(B) = 5 and n(A nn B) =3` then `n[(A xx B) nn (B xx A)]=`

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the problem, we need to find \( n[(A \times B) \cap (B \times A)] \) given the following:

- \( n(A) = 4 \)
- \( n(B) = 5 \)
- \( n(A \cap B) = 3 \)

### Step-by-Step Solution:

1. **Understanding the Sets**:
- Let \( A = \{ a_1, a_2, a_3, a_4 \} \) (4 elements)
- Let \( B = \{ b_1, b_2, b_3, b_4, b_5 \} \) (5 elements)
- The intersection \( A \cap B \) contains 3 common elements, say \( \{ a_1, a_2, a_3 \} \).

2. **Identifying Elements in Sets**:
- The elements in \( A \) can be represented as:
- \( A = \{ a_1, a_2, a_3, a_4 \} \)
- The elements in \( B \) can be represented as:
- \( B = \{ a_1, a_2, a_3, b_4, b_5 \} \) (where \( b_4 \) and \( b_5 \) are not in \( A \)).

3. **Calculating \( n(A \times B) \)**:
- The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from \( A \) and the second is from \( B \).
- Therefore, \( n(A \times B) = n(A) \times n(B) = 4 \times 5 = 20 \).

4. **Calculating \( n(B \times A) \)**:
- Similarly, \( n(B \times A) = n(B) \times n(A) = 5 \times 4 = 20 \).

5. **Finding \( n[(A \times B) \cap (B \times A)] \)**:
- The intersection \( (A \times B) \cap (B \times A) \) consists of pairs \( (x, y) \) such that \( x \in A \) and \( y \in B \) and also \( x \in B \) and \( y \in A \).
- This means both \( x \) and \( y \) must be from the intersection \( A \cap B \).
- The number of such pairs is given by the square of the number of elements in the intersection:
\[ n[(A \times B) \cap (B \times A)] = n(A \cap B)^2 = 3^2 = 9. \]

### Final Answer:
\[ n[(A \times B) \cap (B \times A)] = 9. \]