To define the relation \( R \) from set \( A \) to set \( B \) where the difference between \( x \) and \( y \) is odd, 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: Understand the condition for the relation
The relation \( R \) is defined as:
\[ R = \{(x, y) : |x - y| \text{ is odd}, x \in A, y \in B\} \]
This means we need to find pairs \( (x, y) \) such that the absolute difference \( |x - y| \) is an odd number.
### Step 3: Calculate the differences
We will check each combination of \( x \) from set \( A \) and \( y \) from set \( B \) to see if the difference is odd.
1. For \( x = 1 \):
- \( y = 4 \): \( |1 - 4| = 3 \) (odd) → Include \( (1, 4) \)
- \( y = 6 \): \( |1 - 6| = 5 \) (odd) → Include \( (1, 6) \)
- \( y = 9 \): \( |1 - 9| = 8 \) (even) → Exclude
2. For \( x = 2 \):
- \( y = 4 \): \( |2 - 4| = 2 \) (even) → Exclude
- \( y = 6 \): \( |2 - 6| = 4 \) (even) → Exclude
- \( y = 9 \): \( |2 - 9| = 7 \) (odd) → Include \( (2, 9) \)
3. For \( x = 3 \):
- \( y = 4 \): \( |3 - 4| = 1 \) (odd) → Include \( (3, 4) \)
- \( y = 6 \): \( |3 - 6| = 3 \) (odd) → Include \( (3, 6) \)
- \( y = 9 \): \( |3 - 9| = 6 \) (even) → Exclude
4. For \( x = 5 \):
- \( y = 4 \): \( |5 - 4| = 1 \) (odd) → Include \( (5, 4) \)
- \( y = 6 \): \( |5 - 6| = 1 \) (odd) → Include \( (5, 6) \)
- \( y = 9 \): \( |5 - 9| = 4 \) (even) → Exclude
### Step 4: Compile the ordered pairs
From the calculations, we have the following ordered pairs:
- From \( x = 1 \): \( (1, 4), (1, 6) \)
- From \( x = 2 \): \( (2, 9) \)
- From \( x = 3 \): \( (3, 4), (3, 6) \)
- From \( x = 5 \): \( (5, 4), (5, 6) \)
### Step 5: Write the relation in roster form
Now we can write the relation \( R \) in roster form:
\[ R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\} \]
