Let A = {a,b,c} and B = {4, 5}. Consider a relation R defined from set A to set B, then R can be equal to set

Options

A. A
B. B
C. `AxxB`
D. `BxxA`

Answer

To solve the problem, we need to analyze the sets and the relation defined from set A to set B.

### Step-by-Step Solution:

1. **Identify the Sets:**
- Let \( A = \{ a, b, c \} \)
- Let \( B = \{ 4, 5 \} \)

2. **Define the Relation \( R \):**
- A relation \( R \) from set \( A \) to set \( B \) consists of ordered pairs \( (x, y) \) where \( x \) belongs to \( A \) and \( y \) belongs to \( B \).

3. **Determine Possible Ordered Pairs:**
- The possible ordered pairs can be formed by pairing each element of \( A \) with each element of \( B \):
- From \( a \): \( (a, 4) \), \( (a, 5) \)
- From \( b \): \( (b, 4) \), \( (b, 5) \)
- From \( c \): \( (c, 4) \), \( (c, 5) \)
- Therefore, the complete set of ordered pairs that can form the relation \( R \) is:
\[ R = \{ (a, 4), (a, 5), (b, 4), (b, 5), (c, 4), (c, 5) \} \]

4. **Check the Cartesian Product \( A \times B \):**
- The Cartesian product \( A \times B \) is defined as:
\[ A \times B = \{ (x, y) | x \in A, y \in B \} \]
- Thus, \( A \times B \) is exactly the set of ordered pairs we listed above:
\[ A \times B = \{ (a, 4), (a, 5), (b, 4), (b, 5), (c, 4), (c, 5) \} \]

5. **Check the Cartesian Product \( B \times A \):**
- The Cartesian product \( B \times A \) is defined as:
\[ B \times A = \{ (x, y) | x \in B, y \in A \} \]
- This would yield:
\[ B \times A = \{ (4, a), (4, b), (4, c), (5, a), (5, b), (5, c) \} \]
- None of these pairs are in the relation \( R \) defined from \( A \) to \( B \).

6. **Conclusion:**
- Therefore, the relation \( R \) can be equal to the Cartesian product \( A \times B \) but not to \( B \times A \).
- The final answer is that \( R \) can be equal to \( A \times B \).