If the set A has p elements, B has. q elements, then the number of elements in A `xx B ` is

Options

A. p + q
B. p + q + 1
C. pq
D. `p^(2)`

Answer

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

### Step-by-Step Solution:

1. **Understand the Sets**: Let set A have \( p \) elements and set B have \( q \) elements.

2. **Definition of Cartesian Product**: The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \) is an element from set A and \( b \) is an element from set B.

3. **Forming Ordered Pairs**: For each element in set A, we can pair it with every element in set B. This means:
- The first element of set A can be paired with all \( q \) elements of set B.
- The second element of set A can also be paired with all \( q \) elements of set B.
- This continues for all \( p \) elements in set A.

4. **Counting the Ordered Pairs**: Since each of the \( p \) elements in set A can be paired with \( q \) elements in set B, the total number of ordered pairs (elements in \( A \times B \)) is given by:
\[ \text{Total elements in } A \times B = p \times q \]

5. **Conclusion**: Therefore, the number of elements in the Cartesian product \( A \times B \) is \( p \times q \).

### Final Answer:
The number of elements in \( A \times B \) is \( p \times q \).

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