Let `X = {1, 2, 3, 4}` and `Y = {1, 3, 5, 7,9}`. Which of the following is relations from X to Y if R= { (1,1), (2,1), (3,3), (4, 3), (5,5)}

Options

A. `R_(1) = {(x, y):y=2+x,x inX, yinY}`
B. `R_(2) = {(1, 1), (2, 1), (3, 3), (4, 3), (5, 5)}`
C. `R_(3) = {(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)}`
D. `R_(4) = {(1, 3), (2, 5), (2, 4), (7, 9)}`

Explanation

To determine if the relation \( R = \{ (1,1), (2,1), (3,3), (4,3), (5,5) \} \) is a valid relation from set \( X \) to set \( Y \), we need to check whether all the first elements of the ordered pairs in \( R \) belong to set \( X \) and whether the second elements belong to set \( Y \).

### Step-by-Step Solution:

1. **Identify the Sets:**
- Set \( X = \{1, 2, 3, 4\} \)
- Set \( Y = \{1, 3, 5, 7, 9\} \)

2. **List the Ordered Pairs in Relation \( R \):**
- The relation \( R \) consists of the following pairs:
- \( (1, 1) \)
- \( (2, 1) \)
- \( (3, 3) \)
- \( (4, 3) \)
- \( (5, 5) \)

3. **Check Each Ordered Pair:**
- For \( (1, 1) \):
- First element \( 1 \) is in \( X \) and second element \( 1 \) is in \( Y \) → Valid.
- For \( (2, 1) \):
- First element \( 2 \) is in \( X \) and second element \( 1 \) is in \( Y \) → Valid.
- For \( (3, 3) \):
- First element \( 3 \) is in \( X \) and second element \( 3 \) is in \( Y \) → Valid.
- For \( (4, 3) \):
- First element \( 4 \) is in \( X \) and second element \( 3 \) is in \( Y \) → Valid.
- For \( (5, 5) \):
- First element \( 5 \) is **not** in \( X \) (since \( X \) only contains \( 1, 2, 3, 4 \)) → Invalid.

4. **Conclusion:**
- Since the pair \( (5, 5) \) contains a first element that is not in set \( X \), the relation \( R \) cannot be considered a valid relation from \( X \) to \( Y \).

### Final Answer:
The relation \( R \) is **not** a valid relation from \( X \) to \( Y \).