If each one of the two sets A and B has 3 elements, how many elements are there in `AxxB?` If three of the elements in `AxxB` be (2,3),(4,4) and (6,6), find the remaining elements.

Answer

To solve the problem step by step, we will follow the instructions provided in the question.

### Step 1: Determine the number of elements in A x B
Given that both sets A and B have 3 elements each, we can denote the number of elements in set A as \( n(A) = 3 \) and the number of elements in set B as \( n(B) = 3 \).

The formula for the number of elements in the Cartesian product \( A \times B \) is given by:
\[ n(A \times B) = n(A) \times n(B) \]
Substituting the values:
\[ n(A \times B) = 3 \times 3 = 9 \]
Thus, there are 9 elements in \( A \times B \).

### Step 2: Identify the given elements in A x B
The problem states that three of the elements in \( A \times B \) are:
- (2, 3)
- (4, 4)
- (6, 6)

### Step 3: Determine the elements of sets A and B
From the given elements, we can infer the possible elements of sets A and B.

1. The first components of the pairs (2, 3), (4, 4), and (6, 6) suggest that the elements of set A could be {2, 4, 6}.
2. The second components of the pairs suggest that the elements of set B could be {3, 4, 6}.

### Step 4: List all possible pairs in A x B
Now, we will list all possible pairs that can be formed using the elements of sets A and B.

- For A = {2, 4, 6} and B = {3, 4, 6}, the pairs in \( A \times B \) are:
- (2, 3)
- (2, 4)
- (2, 6)
- (4, 3)
- (4, 4)
- (4, 6)
- (6, 3)
- (6, 4)
- (6, 6)

### Step 5: Identify the remaining elements
From the pairs we listed, we already have three pairs given in the problem. The remaining pairs are:
- (2, 4)
- (2, 6)
- (4, 3)
- (4, 6)
- (6, 3)
- (6, 4)

Thus, the remaining elements in \( A \times B \) are:
- (2, 4)
- (2, 6)
- (4, 3)
- (4, 6)
- (6, 3)
- (6, 4)

### Summary of the solution:
The total number of elements in \( A \times B \) is 9. The three given elements are (2, 3), (4, 4), and (6, 6). The remaining elements are (2, 4), (2, 6), (4, 3), (4, 6), (6, 3), and (6, 4).