If A and B are finite sets (non-empty), then number of elements in `A xxB` is

Options

A. `n (A ∪ B)`
B. `n (A nn B)`
C. `n (A)xx n(B)`
D. none of these

Answer

To determine the number of elements in the Cartesian product of two finite sets A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Sets**: Let A and B be two finite sets. For example, let A = {a1, a2} and B = {b1, b2}.

2. **Count the Elements in Each Set**:
- Count the number of elements in set A. Let's say A has `m` elements.
- Count the number of elements in set B. Let's say B has `n` elements.
- In our example, A has 2 elements (a1, a2) and B has 2 elements (b1, b2).

3. **Understand the Cartesian Product**: The Cartesian product A × B consists of all possible ordered pairs (a, b) where `a` is an element from set A and `b` is an element from set B.

4. **Calculate the Number of Ordered Pairs**: The total number of ordered pairs in A × B is given by the formula:
\[ |A \times B| = |A| \times |B| \]
where |A| is the number of elements in set A and |B| is the number of elements in set B.

5. **Apply the Formula**: Using our example, since |A| = 2 and |B| = 2, we can calculate:
\[ |A \times B| = 2 \times 2 = 4 \]
Thus, the number of elements in A × B is 4.

6. **Conclusion**: Therefore, for any finite sets A and B, the number of elements in the Cartesian product A × B is equal to the product of the number of elements in each set.

### Final Answer:
The number of elements in A × B is given by:
\[ |A \times B| = |A| \times |B| \]