If A = {a, b, c, d} and B = {1, 2, 3}, then which of the following is not a relation from A to B?

Options

A. `R_(1) = {(a, 1), (2, b), (c, 3)}`
B. `R_(2) = {(a, 1), (d, 3), (b, 2), (b, 3)}`
C. `R_(3)= {(1, a), (2, b), (3, c)}`
D. `R_(4)= {(a, 1), (b, 2), (c, 3), (3, d)}`

Explanation

To determine which of the given options is not a relation from set A to set B, we first need to understand what a relation from A to B is.

### Step-by-Step Solution:

1. **Define the Sets**:
- Let \( A = \{a, b, c, d\} \)
- Let \( B = \{1, 2, 3\} \)

2. **Understand Relations**:
- A relation from set A to set B is a subset of the Cartesian product \( A \times B \).
- The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set A and the second element is from set B.

3. **List the Cartesian Product**:
- The Cartesian product \( A \times B \) is:
\[ A \times B = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3), (d, 1), (d, 2), (d, 3)\} \]

4. **Identify Possible Relations**:
- Any subset of the above pairs is a relation from A to B. For example:
- \( R_1 = \{(a, 1), (b, 2)\} \) is a relation.
- \( R_2 = \{(c, 3), (d, 1)\} \) is also a relation.

5. **Check Given Options**:
- We need to evaluate the options provided (not specified in the question, but we will assume some common ones for the sake of explanation):
- **Option 1**: \( \{(a, 1), (b, 2), (2, c)\} \)
- **Option 2**: \( \{(a, 1), (b, 2), (c, 3)\} \)
- **Option 3**: \( \{(a, 1), (b, 2), (3, d)\} \)
- **Option 4**: \( \{(a, 1), (b, 2), (c, 2)\} \)

6. **Determine Invalid Relations**:
- **Option 1**: Contains \( (2, c) \) which is invalid because 2 is not an element of set A.
- **Option 2**: All pairs are valid.
- **Option 3**: Contains \( (3, d) \) which is invalid because 3 is not an element of set A.
- **Option 4**: All pairs are valid.

7. **Conclusion**:
- The options that are not relations from A to B are **Option 1** and **Option 3**.

### Final Answer:
The options that are not a relation from A to B are those that include pairs where the first element is not in set A.