If `A= {2, 3, 5, 7}` and B= {2, 4, 6}. Then, the elements in `A xx B` is

Options

A. 12
B. 11
C. 10
D. 13

Answer

To find the elements in the Cartesian product \( A \times B \) where \( A = \{2, 3, 5, 7\} \) and \( B = \{2, 4, 6\} \), we can follow these steps:

### Step 1: Understand the Cartesian Product
The Cartesian product \( A \times B \) consists of all possible ordered pairs \( (a, b) \) where \( a \) is an element from set \( A \) and \( b \) is an element from set \( B \).

### Step 2: Count the Elements in Each Set
- The number of elements in set \( A \) is 4 (elements: 2, 3, 5, 7).
- The number of elements in set \( B \) is 3 (elements: 2, 4, 6).

### Step 3: Use the Formula for Cartesian Product
The number of elements in the Cartesian product \( A \times B \) can be calculated using 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 \( B \).

### Step 4: Calculate the Number of Elements
Substituting the values we have:
\[ |A \times B| = 4 \times 3 = 12 \]

### Step 5: List the Elements of \( A \times B \)
Now, we can list the ordered pairs:
- From element 2 in \( A \): \( (2, 2), (2, 4), (2, 6) \)
- From element 3 in \( A \): \( (3, 2), (3, 4), (3, 6) \)
- From element 5 in \( A \): \( (5, 2), (5, 4), (5, 6) \)
- From element 7 in \( A \): \( (7, 2), (7, 4), (7, 6) \)

Combining all these pairs, we have:
\[ A \times B = \{(2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6), (7, 2), (7, 4), (7, 6)\} \]

### Final Result
Thus, the number of elements in \( A \times B \) is 12, and the elements are:
\[ \{(2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6), (7, 2), (7, 4), (7, 6)\} \]