To define the relation \( R \) from set \( A \) to set \( B \) where \( R = \{(x, y) : |x - y| \text{ is odd}, x \in A, y \in B\} \), we will follow these steps:
### Step 1: Identify the sets
We have:
- Set \( A = \{1, 2, 3, 5\} \)
- Set \( B = \{4, 6, 9\} \)
### Step 2: Determine the conditions for the relation
The relation \( R \) consists of ordered pairs \( (x, y) \) such that the difference \( |x - y| \) is odd.
### Step 3: List all possible ordered pairs from \( A \) to \( B \)
The Cartesian product \( A \times B \) includes all possible pairs:
- \( (1, 4) \)
- \( (1, 6) \)
- \( (1, 9) \)
- \( (2, 4) \)
- \( (2, 6) \)
- \( (2, 9) \)
- \( (3, 4) \)
- \( (3, 6) \)
- \( (3, 9) \)
- \( (5, 4) \)
- \( (5, 6) \)
- \( (5, 9) \)
### Step 4: Check each pair to see if the difference is odd
1. \( (1, 4) \): \( |1 - 4| = 3 \) (odd) → Include
2. \( (1, 6) \): \( |1 - 6| = 5 \) (odd) → Include
3. \( (1, 9) \): \( |1 - 9| = 8 \) (even) → Exclude
4. \( (2, 4) \): \( |2 - 4| = 2 \) (even) → Exclude
5. \( (2, 6) \): \( |2 - 6| = 4 \) (even) → Exclude
6. \( (2, 9) \): \( |2 - 9| = 7 \) (odd) → Include
7. \( (3, 4) \): \( |3 - 4| = 1 \) (odd) → Include
8. \( (3, 6) \): \( |3 - 6| = 3 \) (odd) → Include
9. \( (3, 9) \): \( |3 - 9| = 6 \) (even) → Exclude
10. \( (5, 4) \): \( |5 - 4| = 1 \) (odd) → Include
11. \( (5, 6) \): \( |5 - 6| = 1 \) (odd) → Include
12. \( (5, 9) \): \( |5 - 9| = 4 \) (even) → Exclude
### Step 5: Compile the relation \( R \)
The relation \( R \) in roster form is:
\[
R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\}
\]
