If `A={1,2,3,4},B={5,6,7,8}`, then which of the following are relations from A to B? Give reasons for your answer. `R_(1)={(1,5),(2,7),(3,8)}`

Answer

To determine whether the set \( R_1 = \{(1,5), (2,7), (3,8)\} \) is a relation from set \( A \) to set \( B \), we need to follow these steps:

### Step 1: Define the Sets
Identify the elements of the sets:
- \( A = \{1, 2, 3, 4\} \)
- \( B = \{5, 6, 7, 8\} \)

### Step 2: Determine the Cartesian Product \( A \times B \)
The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from set \( A \) and the second element is from set \( B \). We can list these pairs:

- For \( 1 \) from \( A \): \( (1,5), (1,6), (1,7), (1,8) \)
- For \( 2 \) from \( A \): \( (2,5), (2,6), (2,7), (2,8) \)
- For \( 3 \) from \( A \): \( (3,5), (3,6), (3,7), (3,8) \)
- For \( 4 \) from \( A \): \( (4,5), (4,6), (4,7), (4,8) \)

Thus, the Cartesian product \( A \times B \) is:
\[ A \times B = \{(1,5), (1,6), (1,7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8)\} \]

### Step 3: Check if \( R_1 \) is a Subset of \( A \times B \)
Now, we need to check if each element of \( R_1 \) is present in \( A \times B \):
- \( (1,5) \) is in \( A \times B \)
- \( (2,7) \) is in \( A \times B \)
- \( (3,8) \) is in \( A \times B \)

Since all pairs in \( R_1 \) are found in \( A \times B \), we conclude that \( R_1 \) is indeed a subset of \( A \times B \).

### Conclusion
Therefore, \( R_1 \) is a relation from set \( A \) to set \( B \).

---