Let A and B be two sets such that `AxxB` consists of 6 elements. If three elements of `AxxB` are: `(1,4),\ (2,6),\ (3,6)` . Find `A\ a n d\ Bdot`

Options

A. A = {1, 2} and B = {3, 4, 6}
B. A = {4, 6} and B = {1, 2, 3}
C. A = {1, 2, 3} and B = {4, 6}
D. A = {1, 2, 4} and B = {3, 6}

Explanation

To solve the problem, we need to find the sets A and B given that the Cartesian product \( A \times B \) consists of 6 elements, and three of those elements are \( (1, 4), (2, 6), (3, 6) \).

### Step-by-Step Solution:

1. **Understanding the Cartesian Product**:
The Cartesian product \( A \times B \) is defined as the set of all ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \). The number of elements in \( A \times B \) is given by the formula:
\[ |A \times B| = |A| \times |B| \]
where \( |A| \) is the number of elements in set A and \( |B| \) is the number of elements in set B.

2. **Given Information**:
We know that \( |A \times B| = 6 \). This means:
\[ |A| \times |B| = 6 \]

3. **Finding Possible Sizes for A and B**:
The pairs of positive integers that multiply to 6 are:
- \( (1, 6) \)
- \( (2, 3) \)
- \( (3, 2) \)
- \( (6, 1) \)

Since we need to find sets A and B, we can eliminate \( (1, 6) \) and \( (6, 1) \) because they would imply one of the sets is empty, which is not possible.

4. **Choosing Sizes for A and B**:
The feasible pairs are \( (2, 3) \) or \( (3, 2) \). This means:
- If \( |A| = 2 \) and \( |B| = 3 \), or
- If \( |A| = 3 \) and \( |B| = 2 \).

5. **Identifying Elements in A and B**:
From the given pairs \( (1, 4), (2, 6), (3, 6) \):
- The first elements (1, 2, 3) suggest that \( A = \{1, 2, 3\} \).
- The second elements (4, 6) suggest that \( B = \{4, 6\} \).

6. **Verifying the Sizes**:
- If \( A = \{1, 2, 3\} \) then \( |A| = 3 \).
- If \( B = \{4, 6\} \) then \( |B| = 2 \).
- Therefore, \( |A \times B| = 3 \times 2 = 6 \), which matches the given information.

### Conclusion:
Thus, the sets are:
- \( A = \{1, 2, 3\} \)
- \( B = \{4, 6\} \)