Let `X={1,2,3,4,5}` and `Y={1,3,5,7,9}`. Which of the following is not a relation from `X` to `Y`

Options

A. `R_(1)={(x,y)|y=2+x,xepsilonX,yepsilonY}`
B. `R_(2)={(1,1),(2,1),(3,3),(4,4),(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 which of the given relations is not a relation from set \( X \) to set \( Y \), we need to analyze each relation based on the definitions of the sets.

Given:
- \( X = \{1, 2, 3, 4, 5\} \)
- \( Y = \{1, 3, 5, 7, 9\} \)

### Step-by-Step Solution:

1. **Identify the relations**: We are given four relations (let's denote them as \( R_1, R_2, R_3, R_4 \)). We need to check each relation to see if it contains pairs where the first element is from \( X \) and the second element is from \( Y \).

2. **Check \( R_1 \)**:
- \( R_1 \) is defined as \( y = 2 + x \).
- For \( x = 1 \): \( y = 2 + 1 = 3 \) (valid, since 3 is in \( Y \))
- For \( x = 2 \): \( y = 2 + 2 = 4 \) (invalid, since 4 is not in \( Y \))
- Since for \( x = 2 \), \( y \) is not in \( Y \), \( R_1 \) is not a valid relation from \( X \) to \( Y \).

3. **Check \( R_2 \)**:
- \( R_2 = \{(1, 1), (2, 1), (3, 3), (4, 4), (5, 5)\} \)
- All pairs have their first elements in \( X \) and second elements in \( Y \).
- Thus, \( R_2 \) is a valid relation.

4. **Check \( R_3 \)**:
- \( R_3 = \{(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)\} \)
- The pairs \( (1, 1), (1, 3), (3, 5) \) are valid since their first elements are in \( X \) and second elements are in \( Y \).
- However, \( (3, 7) \) and \( (5, 7) \) are valid as well since 7 is in \( Y \).
- Thus, \( R_3 \) is a valid relation.

5. **Check \( R_4 \)**:
- \( R_4 = \{(1, 3), (2, 5), (2, 4), (7, 9)\} \)
- The pairs \( (1, 3) \) and \( (2, 5) \) are valid.
- However, \( (2, 4) \) is invalid since 4 is not in \( Y \).
- The pair \( (7, 9) \) is also invalid since 7 is not in \( X \).
- Thus, \( R_4 \) is not a valid relation.

### Conclusion:
The relations \( R_1 \) and \( R_4 \) are not relations from \( X \) to \( Y \). However, since the question asks for which of the following is not a relation, we can conclude that:

- **Answer**: \( R_1 \) and \( R_4 \) are not relations from \( X \) to \( Y \).