A = { 1,2,3,5} and B = {4,6,9} A relation R is defined from A to B by R = { (x,y) : the difference between x & y is odd}. Writer R in roster form.

Explanation

To find the relation \( R \) from set \( A \) to set \( B \) defined by the condition that the difference between \( x \) (from set \( A \)) and \( y \) (from set \( B \)) is odd, we can 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 such that \( R = \{ (x, y) : x \in A, y \in B \text{ and } |x - y| \text{ is odd} \} \). This means we need to find pairs \( (x, y) \) where the absolute difference \( |x - y| \) is an odd number.

### Step 3: Check each combination of elements from \( A \) and \( B \)
We will check each element \( x \) from set \( A \) against each element \( y \) from set \( B \):

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) → do not include

2. For \( x = 2 \):
- \( y = 4 \): \( |2 - 4| = 2 \) (even) → do not include
- \( y = 6 \): \( |2 - 6| = 4 \) (even) → do not include
- \( y = 9 \): \( |2 - 9| = 7 \) (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) → do not include

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) → do not include

### Step 4: Compile the relation in roster form
From our checks, we have the following pairs:
- \( (1, 4) \)
- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 4) \)
- \( (3, 6) \)
- \( (5, 4) \)
- \( (5, 6) \)

Thus, the relation \( R \) in roster form is:
\[ R = \{ (1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6) \} \]