To solve the problem step by step, we need to find the relation \( R \) defined by the condition that \( a - b \) is odd for \( a \in A \) and \( b \in B \).
### Step 1: Identify the sets
We have:
- Set \( A = \{1, 2, 3, 5\} \)
- Set \( B = \{4, 6, 9\} \)
### Step 2: Determine the pairs \((a, b)\) such that \( a - b \) is odd
To find the pairs, we will check each element of set \( A \) with each element of set \( B \):
1. For \( a = 1 \):
- \( 1 - 4 = -3 \) (odd) → include (1, 4)
- \( 1 - 6 = -5 \) (odd) → include (1, 6)
- \( 1 - 9 = -8 \) (even) → do not include
2. For \( a = 2 \):
- \( 2 - 4 = -2 \) (even) → do not include
- \( 2 - 6 = -4 \) (even) → do not include
- \( 2 - 9 = -7 \) (odd) → include (2, 9)
3. For \( a = 3 \):
- \( 3 - 4 = -1 \) (odd) → include (3, 4)
- \( 3 - 6 = -3 \) (odd) → include (3, 6)
- \( 3 - 9 = -6 \) (even) → do not include
4. For \( a = 5 \):
- \( 5 - 4 = 1 \) (odd) → include (5, 4)
- \( 5 - 6 = -1 \) (odd) → include (5, 6)
- \( 5 - 9 = -4 \) (even) → do not include
### Step 3: Compile the pairs into roster form
From the calculations above, the pairs that satisfy the condition \( a - b \) is odd are:
- \( (1, 4) \)
- \( (1, 6) \)
- \( (2, 9) \)
- \( (3, 4) \)
- \( (3, 6) \)
- \( (5, 4) \)
- \( (5, 6) \)
Thus, the relation \( R \) in roster form is:
\[
R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\}
\]
### Step 4: Represent \( R \) by an arrow diagram
To represent the relation \( R \) using an arrow diagram:
- Draw two circles, one for set \( A \) and one for set \( B \).
- Write the elements of set \( A \) inside the first circle: \( 1, 2, 3, 5 \).
- Write the elements of set \( B \) inside the second circle: \( 4, 6, 9 \).
- Draw arrows from elements of \( A \) to elements of \( B \) based on the pairs we found:
- From \( 1 \) to \( 4 \) and \( 6 \)
- From \( 2 \) to \( 9 \)
- From \( 3 \) to \( 4 \) and \( 6 \)
- From \( 5 \) to \( 4 \) and \( 6 \)
