Let `A={1 , 3,5,7}` and `B={2,4,6,8}` be two sets and let `R` be a relation from `A` to `B` defined by the phrase `(x,y) in R harr x gt y` Find the number of ordered pairs under this relation `R`,

Options

A. 8
B. 10
C. 6
D. 4

Answer

To find the number of ordered pairs under the relation \( R \) defined by \( (x, y) \in R \) if \( x > y \), we will analyze the sets \( A \) and \( B \) given as follows:

- Set \( A = \{1, 3, 5, 7\} \)
- Set \( B = \{2, 4, 6, 8\} \)

### Step 1: Identify the elements of sets \( A \) and \( B \)
We have:
- Elements of \( A \): 1, 3, 5, 7
- Elements of \( B \): 2, 4, 6, 8

### Step 2: Determine the ordered pairs \( (x, y) \) such that \( x > y \)
We will check each element of set \( A \) against all elements of set \( B \) to see if the condition \( x > y \) holds.

1. **For \( x = 1 \)**:
- \( 1 > 2 \) → False
- \( 1 > 4 \) → False
- \( 1 > 6 \) → False
- \( 1 > 8 \) → False
- **No valid pairs.**

2. **For \( x = 3 \)**:
- \( 3 > 2 \) → True → Pair: \( (3, 2) \)
- \( 3 > 4 \) → False
- \( 3 > 6 \) → False
- \( 3 > 8 \) → False
- **Valid pairs: 1 (i.e., \( (3, 2) \)).**

3. **For \( x = 5 \)**:
- \( 5 > 2 \) → True → Pair: \( (5, 2) \)
- \( 5 > 4 \) → True → Pair: \( (5, 4) \)
- \( 5 > 6 \) → False
- \( 5 > 8 \) → False
- **Valid pairs: 2 (i.e., \( (5, 2), (5, 4) \)).**

4. **For \( x = 7 \)**:
- \( 7 > 2 \) → True → Pair: \( (7, 2) \)
- \( 7 > 4 \) → True → Pair: \( (7, 4) \)
- \( 7 > 6 \) → True → Pair: \( (7, 6) \)
- \( 7 > 8 \) → False
- **Valid pairs: 3 (i.e., \( (7, 2), (7, 4), (7, 6) \)).**

### Step 3: Count the total number of valid ordered pairs
Now we will sum the valid pairs from each case:
- From \( x = 1 \): 0 pairs
- From \( x = 3 \): 1 pair
- From \( x = 5 \): 2 pairs
- From \( x = 7 \): 3 pairs

Total valid pairs = \( 0 + 1 + 2 + 3 = 6 \)

### Conclusion
The total number of ordered pairs \( (x, y) \) such that \( x > y \) is **6**.