The Certesian product `AxxA` has 16 elements `S={(a,b)epsilonAxxA|altb}.(-1,2)` and `(0,1)` are two elements belonging to `S`. Find the set containing the remaining elements of `S`.

Explanation

To solve the problem, we need to find the remaining elements of the set \( S \) from the Cartesian product \( A \times A \) given the conditions. Let's break it down step by step.

### Step 1: Identify the elements of set \( A \)

From the elements provided in set \( S \), we have:
- The elements \( (-1, 2) \) and \( (0, 1) \) belong to \( S \).
- The condition states that \( a < b \).

From the pairs \( (-1, 2) \) and \( (0, 1) \), we can derive the elements of set \( A \):
- The first element of the first pair is \( -1 \).
- The second element of the first pair is \( 2 \).
- The first element of the second pair is \( 0 \).
- The second element of the second pair is \( 1 \).

Thus, the set \( A \) can be identified as:
\[ A = \{-1, 0, 1, 2\} \]

### Step 2: Determine the Cartesian product \( A \times A \)

Next, we compute the Cartesian product \( A \times A \):
\[ A \times A = \{(x, y) | x \in A, y \in A\} \]
This results in the following pairs:
- \( (-1, -1), (-1, 0), (-1, 1), (-1, 2) \)
- \( (0, -1), (0, 0), (0, 1), (0, 2) \)
- \( (1, -1), (1, 0), (1, 1), (1, 2) \)
- \( (2, -1), (2, 0), (2, 1), (2, 2) \)

### Step 3: Filter the pairs based on the condition \( a < b \)

Now, we filter these pairs to satisfy the condition \( a < b \):
- From \( (-1, -1) \): Not included (not \( a < b \))
- From \( (-1, 0) \): Included
- From \( (-1, 1) \): Included
- From \( (-1, 2) \): Included
- From \( (0, -1) \): Not included (not \( a < b \))
- From \( (0, 0) \): Not included (not \( a < b \))
- From \( (0, 1) \): Included
- From \( (0, 2) \): Included
- From \( (1, -1) \): Not included (not \( a < b \))
- From \( (1, 0) \): Not included (not \( a < b \))
- From \( (1, 1) \): Not included (not \( a < b \))
- From \( (1, 2) \): Included
- From \( (2, -1) \): Not included (not \( a < b \))
- From \( (2, 0) \): Not included (not \( a < b \))
- From \( (2, 1) \): Not included (not \( a < b \))
- From \( (2, 2) \): Not included (not \( a < b \))

### Step 4: Compile the remaining elements of set \( S \)

The pairs that satisfy \( a < b \) are:
- \( (-1, 0) \)
- \( (-1, 1) \)
- \( (-1, 2) \)
- \( (0, 1) \)
- \( (0, 2) \)
- \( (1, 2) \)

Now, we already know that \( (-1, 2) \) and \( (0, 1) \) are included in \( S \). Therefore, the remaining elements of set \( S \) are:
\[ \{(-1, 0), (-1, 1), (0, 2), (1, 2)\} \]

### Final Answer

The set containing the remaining elements of \( S \) is:
\[ \{(-1, 0), (-1, 1), (0, 2), (1, 2)\} \]