A and B are two sets given in such a way that `A xx B` contains 6 elements. If three elements of `A xx B` be `(1, 3), (2, 5) and (3,3)`. Then, the remaining elements are

Options

A. `(1, 3), (2, 3) and (3, 5)`
B. `(2, 3), (3,1) and (4, 5)`
C. `(2, 3), (3, 4) and (4, 5)`
D. `(1, 5), (2, 3) and (3, 5)`

Answer

To solve the problem step by step, we need to find the remaining elements of the Cartesian product \( A \times B \) given that it contains 6 elements in total, and 3 of them are already specified.

### Step-by-Step Solution:

1. **Understand the Cartesian Product**: The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).

2. **Identify Given Information**: We know that:
- \( |A \times B| = 6 \) (total elements in the Cartesian product)
- Three elements of \( A \times B \) are given: \( (1, 3), (2, 5), (3, 3) \).

3. **Count the Given Elements**: We have 3 elements already provided. Therefore, the number of remaining elements is:
\[ 6 - 3 = 3 \]

4. **Determine Possible Elements**: We need to find the pairs that can be formed from the elements of sets \( A \) and \( B \).
- Let's denote the elements of set \( A \) as \( \{1, 2, 3\} \) and the elements of set \( B \) as \( \{3, 5\} \) (since these are the second elements of the given pairs).

5. **List All Possible Pairs**: The possible pairs from \( A \) and \( B \) are:
- From \( 1 \): \( (1, 3), (1, 5) \)
- From \( 2 \): \( (2, 3), (2, 5) \)
- From \( 3 \): \( (3, 3), (3, 5) \)

So, the complete set of pairs is:
\[ (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5) \]

6. **Identify Remaining Elements**: We already have the pairs \( (1, 3), (2, 5), (3, 3) \). The remaining pairs from our list are:
- \( (1, 5) \)
- \( (2, 3) \)
- \( (3, 5) \)

7. **Final Answer**: Thus, the remaining elements of \( A \times B \) are:
\[ (1, 5), (2, 3), (3, 5) \]