A set B is given as `B={1,2}`. Some elements of `AxxB` are (3,1),(5,1) and (7,2) Find the remaining elements `AxxB` of such that `n(AxxB)` is least.

Answer

To solve the problem, we need to find the remaining elements of the Cartesian product \( A \times B \) such that the number of elements \( n(A \times B) \) is minimized. Given the set \( B = \{1, 2\} \) and some elements of \( A \times B \) are \( (3, 1), (5, 1), (7, 2) \), we can follow these steps:

### Step 1: Identify the elements of set B
We know that the set \( B \) is given as:
\[ B = \{1, 2\} \]

### Step 2: List the known elements of \( A \times B \)
The elements provided from \( A \times B \) are:
\[ (3, 1), (5, 1), (7, 2) \]

### Step 3: Determine the elements of set A
From the known pairs, we can extract the first elements, which represent the elements of set \( A \):
- From \( (3, 1) \), we have \( 3 \)
- From \( (5, 1) \), we have \( 5 \)
- From \( (7, 2) \), we have \( 7 \)

Thus, we can conclude:
\[ A = \{3, 5, 7\} \]

### Step 4: Calculate the total number of elements in \( A \times B \)
The Cartesian product \( A \times B \) will consist of all possible pairs formed by elements of \( A \) and \( B \). Since \( A \) has 3 elements and \( B \) has 2 elements, the total number of elements in \( A \times B \) is:
\[ n(A \times B) = n(A) \times n(B) = 3 \times 2 = 6 \]

### Step 5: List all possible pairs in \( A \times B \)
The complete set of pairs in \( A \times B \) is:
- \( (3, 1) \)
- \( (3, 2) \)
- \( (5, 1) \)
- \( (5, 2) \)
- \( (7, 1) \)
- \( (7, 2) \)

### Step 6: Identify the remaining elements
From the pairs we listed, we can see that the following pairs are missing from the ones provided:
- \( (3, 2) \)
- \( (5, 2) \)
- \( (7, 1) \)
- \( (7, 2) \)

However, since \( (7, 1) \) is already included in the known pairs, the remaining pairs that need to be added to complete \( A \times B \) are:
- \( (3, 2) \)
- \( (5, 2) \)
- \( (7, 1) \)

### Final Answer
The remaining elements of \( A \times B \) are:
- \( (3, 2) \)
- \( (5, 2) \)
- \( (7, 1) \)