If A={2,4,6,8} and B={1,3,5}, then find the domain and range of the relation:
`R{(x,y):x in A, y in B and x gt y}`

Answer

To solve the problem, we need to find the relation \( R \) defined by the condition \( (x,y) \) where \( x \in A \), \( y \in B \), and \( x > y \). Here are the steps to find the relation, domain, and range:

### Step 1: Identify the sets
Given:
- \( A = \{2, 4, 6, 8\} \)
- \( B = \{1, 3, 5\} \)

### Step 2: Form the relation \( R \)
We need to find all pairs \( (x, y) \) such that \( x \) is from set \( A \), \( y \) is from set \( B \), and \( x > y \).

1. **For \( x = 2 \)**:
- \( y = 1 \): \( 2 > 1 \) → Include \( (2, 1) \)
- \( y = 3 \): \( 2 > 3 \) → Exclude
- \( y = 5 \): \( 2 > 5 \) → Exclude
- Result: \( (2, 1) \)

2. **For \( x = 4 \)**:
- \( y = 1 \): \( 4 > 1 \) → Include \( (4, 1) \)
- \( y = 3 \): \( 4 > 3 \) → Include \( (4, 3) \)
- \( y = 5 \): \( 4 > 5 \) → Exclude
- Result: \( (4, 1), (4, 3) \)

3. **For \( x = 6 \)**:
- \( y = 1 \): \( 6 > 1 \) → Include \( (6, 1) \)
- \( y = 3 \): \( 6 > 3 \) → Include \( (6, 3) \)
- \( y = 5 \): \( 6 > 5 \) → Include \( (6, 5) \)
- Result: \( (6, 1), (6, 3), (6, 5) \)

4. **For \( x = 8 \)**:
- \( y = 1 \): \( 8 > 1 \) → Include \( (8, 1) \)
- \( y = 3 \): \( 8 > 3 \) → Include \( (8, 3) \)
- \( y = 5 \): \( 8 > 5 \) → Include \( (8, 5) \)
- Result: \( (8, 1), (8, 3), (8, 5) \)

### Step 3: Compile the relation \( R \)
Now, we can compile all the pairs we found:
\[ R = \{(2, 1), (4, 1), (4, 3), (6, 1), (6, 3), (6, 5), (8, 1), (8, 3), (8, 5)\} \]

### Step 4: Find the domain and range
- **Domain**: The set of all first elements \( x \) from the pairs in \( R \):
\[ \text{Domain} = \{2, 4, 6, 8\} \]

- **Range**: The set of all second elements \( y \) from the pairs in \( R \):
\[ \text{Range} = \{1, 3, 5\} \]

### Final Answer
- **Domain**: \( \{2, 4, 6, 8\} \)
- **Range**: \( \{1, 3, 5\} \)