The ordered pairs (1,1),(2,2),(3,3) are among the elements in the set `AxxB`. If A and B have 3 elements each, how many elements in all does the set `AxxB` have? Also find the remaining elements.

Answer

To solve the problem step by step, we will determine the total number of elements in the set \( A \times B \) and then find the remaining elements.

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

### Step 2: Determine the Number of Elements in Sets A and B
We are given that both sets \( A \) and \( B \) have 3 elements each. Let's denote:
- \( |A| = 3 \)
- \( |B| = 3 \)

### Step 3: Calculate the Total Number of Elements in \( A \times B \)
The total number of elements in the Cartesian product \( A \times B \) can be calculated using the formula:
\[ |A \times B| = |A| \times |B| \]
Substituting the values we have:
\[ |A \times B| = 3 \times 3 = 9 \]

### Step 4: List the Ordered Pairs in \( A \times B \)
Since both sets \( A \) and \( B \) contain the elements \( 1, 2, 3 \), the ordered pairs in \( A \times B \) will be:
- \( (1, 1) \)
- \( (1, 2) \)
- \( (1, 3) \)
- \( (2, 1) \)
- \( (2, 2) \)
- \( (2, 3) \)
- \( (3, 1) \)
- \( (3, 2) \)
- \( (3, 3) \)

### Step 5: Identify the Given Elements
The problem states that the ordered pairs \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) are already included in the set \( A \times B \).

### Step 6: Find the Remaining Elements
To find the remaining elements, we will exclude the given pairs from the complete list of pairs:
- Given pairs: \( (1, 1) \), \( (2, 2) \), \( (3, 3) \)
- Remaining pairs:
- \( (1, 2) \)
- \( (1, 3) \)
- \( (2, 1) \)
- \( (2, 3) \)
- \( (3, 1) \)
- \( (3, 2) \)

### Conclusion
Thus, the total number of elements in the set \( A \times B \) is 9, and the remaining elements are:
- \( (1, 2) \)
- \( (1, 3) \)
- \( (2, 1) \)
- \( (2, 3) \)
- \( (3, 1) \)
- \( (3, 2) \)