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 options is not a relation from set \( X \) to set \( Y \), we first need to understand the definition of a relation between two sets. A relation \( R \) from set \( X \) to set \( Y \) is defined as a subset of the Cartesian product \( X \times Y \). This means that every ordered pair \( (x, y) \) in the relation must have \( x \) belonging to set \( X \) and \( y \) belonging to set \( Y \).

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

Now, we will analyze each option provided to check if it is a valid relation from \( X \) to \( Y \).

### Step 1: Check Option 1
Let \( R_1 = \{(x, y) \mid y = 2 + x, x \in X, y \in Y\} \).

- For \( x = 1 \), \( y = 2 + 1 = 3 \) → valid (3 ∈ Y)
- For \( x = 2 \), \( y = 2 + 2 = 4 \) → valid (4 ∈ Y)
- For \( x = 3 \), \( y = 2 + 3 = 5 \) → valid (5 ∈ Y)
- For \( x = 4 \), \( y = 2 + 4 = 6 \) → invalid (6 ∉ Y)
- For \( x = 5 \), \( y = 2 + 5 = 7 \) → valid (7 ∈ Y)

Since \( R_1 \) contains the pair \( (4, 6) \) which is not valid, this option is not a relation.

### Step 2: Check Option 2
Let \( R_2 = \{(1, 1), (2, 1), (3, 3), (4, 4)\} \).

- \( (1, 1) \): valid (1 ∈ X, 1 ∈ Y)
- \( (2, 1) \): valid (2 ∈ X, 1 ∈ Y)
- \( (3, 3) \): valid (3 ∈ X, 3 ∈ Y)
- \( (4, 4) \): invalid (4 ∈ X, but 4 ∉ Y)

Since \( (4, 4) \) is not a valid pair, this option is not a relation.

### Step 3: Check Option 3
Let \( R_3 = \{(1, 1), (1, 3), (3, 5), (3, 7), (5, 7)\} \).

- \( (1, 1) \): valid (1 ∈ X, 1 ∈ Y)
- \( (1, 3) \): valid (1 ∈ X, 3 ∈ Y)
- \( (3, 5) \): valid (3 ∈ X, 5 ∈ Y)
- \( (3, 7) \): valid (3 ∈ X, 7 ∈ Y)
- \( (5, 7) \): valid (5 ∈ X, 7 ∈ Y)

All pairs are valid, so this option is a relation.

### Step 4: Check Option 4
Let \( R_4 = \{(1, 3), (2, 5), (2, 4)\} \).

- \( (1, 3) \): valid (1 ∈ X, 3 ∈ Y)
- \( (2, 5) \): valid (2 ∈ X, 5 ∈ Y)
- \( (2, 4) \): invalid (2 ∈ X, but 4 ∉ Y)

Since \( (2, 4) \) is not a valid pair, this option is not a relation.

### Conclusion
The options that are not valid relations from set \( X \) to set \( Y \) are:
- Option 1
- Option 2
- Option 4