Let `A= {2,3,5,7}, B= {1,12,13,15}`. How many elements are there in `A xx B`? In `B xx A`? Is `A xx B = B xx A`? Is `n(A xx B) = n(B xx A)`?

Answer

To solve the problem step by step, we will first define the sets and then calculate the Cartesian products \( A \times B \) and \( B \times A \). After that, we will determine the number of elements in each product and check for equality.

### Step 1: Define the sets
Let:
- \( A = \{2, 3, 5, 7\} \)
- \( B = \{1, 12, 13, 15\} \)

### Step 2: Calculate \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs \( (a, b) \) where \( a \) is from set \( A \) and \( b \) is from set \( B \).

The pairs are:
- For \( a = 2 \): \( (2, 1), (2, 12), (2, 13), (2, 15) \)
- For \( a = 3 \): \( (3, 1), (3, 12), (3, 13), (3, 15) \)
- For \( a = 5 \): \( (5, 1), (5, 12), (5, 13), (5, 15) \)
- For \( a = 7 \): \( (7, 1), (7, 12), (7, 13), (7, 15) \)

Thus, the complete set \( A \times B \) is:
\[ A \times B = \{(2, 1), (2, 12), (2, 13), (2, 15), (3, 1), (3, 12), (3, 13), (3, 15), (5, 1), (5, 12), (5, 13), (5, 15), (7, 1), (7, 12), (7, 13), (7, 15)\} \]

### Step 3: Count the elements in \( A \times B \)
The number of elements in \( A \) is 4 and the number of elements in \( B \) is 4. Therefore, the number of elements in \( A \times B \) is given by:
\[ n(A \times B) = n(A) \times n(B) = 4 \times 4 = 16 \]

### Step 4: Calculate \( B \times A \)
Now, we calculate \( B \times A \). This consists of all ordered pairs \( (b, a) \) where \( b \) is from set \( B \) and \( a \) is from set \( A \).

The pairs are:
- For \( b = 1 \): \( (1, 2), (1, 3), (1, 5), (1, 7) \)
- For \( b = 12 \): \( (12, 2), (12, 3), (12, 5), (12, 7) \)
- For \( b = 13 \): \( (13, 2), (13, 3), (13, 5), (13, 7) \)
- For \( b = 15 \): \( (15, 2), (15, 3), (15, 5), (15, 7) \)

Thus, the complete set \( B \times A \) is:
\[ B \times A = \{(1, 2), (1, 3), (1, 5), (1, 7), (12, 2), (12, 3), (12, 5), (12, 7), (13, 2), (13, 3), (13, 5), (13, 7), (15, 2), (15, 3), (15, 5), (15, 7)\} \]

### Step 5: Count the elements in \( B \times A \)
Similarly, the number of elements in \( B \times A \) is:
\[ n(B \times A) = n(B) \times n(A) = 4 \times 4 = 16 \]

### Step 6: Check for equality \( A \times B = B \times A \)
The sets \( A \times B \) and \( B \times A \) are not equal because the ordered pairs are reversed. For example, \( (2, 1) \) is in \( A \times B \) but \( (1, 2) \) is in \( B \times A \).

### Step 7: Check if \( n(A \times B) = n(B \times A) \)
We found that:
\[ n(A \times B) = 16 \quad \text{and} \quad n(B \times A) = 16 \]
Thus, \( n(A \times B) = n(B \times A) \).

### Final Answers:
1. Number of elements in \( A \times B \) = 16
2. Number of elements in \( B \times A \) = 16
3. \( A \times B \neq B \times A \)
4. \( n(A \times B) = n(B \times A) \)