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

Options

A. `R_(1)={(x,y)|y=2+x,x in X,y in Y}`
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)}`

Answer

To determine which of the given relations are defined from the set \( X = \{1, 2, 3, 4\} \) to the set \( Y = \{1, 3, 5, 7, 9\} \), we need to check if each relation consists of ordered pairs where the first element is from \( X \) and the second element is from \( Y \).

### Step-by-step Solution:

1. **Identify the Relations**: We need to check each relation provided in the options to see if they are valid relations from \( X \) to \( Y \).

2. **Check Relation R1**:
- Let's assume \( R1 \) contains pairs defined such that \( y = 2 + x \).
- For \( x = 1 \): \( y = 2 + 1 = 3 \) → valid pair (1, 3).
- For \( x = 2 \): \( y = 2 + 2 = 4 \) → 4 is not in \( Y \) → invalid pair (2, 4).
- Since one of the pairs is invalid, \( R1 \) is **not a relation from \( X \) to \( Y \)**.

3. **Check Relation R2**:
- Let's assume \( R2 \) contains pairs: (1, 1), (2, 1), (3, 3), (4, 5).
- (1, 1) → valid (1 is in \( X \), 1 is in \( Y \)).
- (2, 1) → valid (2 is in \( X \), 1 is in \( Y \)).
- (3, 3) → valid (3 is in \( X \), 3 is in \( Y \)).
- (4, 5) → valid (4 is in \( X \), 5 is in \( Y \)).
- All pairs are valid, so \( R2 \) **is a relation from \( X \) to \( Y \)**.

4. **Check Relation R3**:
- Let's assume \( R3 \) contains pairs: (1, 1), (1, 3), (3, 5), (3, 7).
- (1, 1) → valid (1 is in \( X \), 1 is in \( Y \)).
- (1, 3) → valid (1 is in \( X \), 3 is in \( Y \)).
- (3, 5) → valid (3 is in \( X \), 5 is in \( Y \)).
- (3, 7) → valid (3 is in \( X \), 7 is in \( Y \)).
- All pairs are valid, so \( R3 \) **is a relation from \( X \) to \( Y \)**.

5. **Check Relation R4**:
- Let's assume \( R4 \) contains pairs: (1, 3), (2, 5), (2, 4).
- (1, 3) → valid (1 is in \( X \), 3 is in \( Y \)).
- (2, 5) → valid (2 is in \( X \), 5 is in \( Y \)).
- (2, 4) → invalid (4 is not in \( Y \)).
- Since one of the pairs is invalid, \( R4 \) is **not a relation from \( X \) to \( Y \)**.

### Conclusion:
- The only relations from \( X \) to \( Y \) are \( R2 \) and \( R3 \).
- Therefore, the answer is that **R2 and R3 are relations from \( X \) to \( Y \)**.