Let `A = {x,y,z} and B = {a,b,c,d}` . Which one of the following is not a relation from A to B

Options

A. `{(x,a), (x,c)}`
B. `{(y,c), (y,d)}`
C. `{(z,a), (z,d)}`
D. `{(z,b), (y,b), (a,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 set A to set B is.

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \( A = \{x, y, z\} \)
- Let \( B = \{a, b, c, d\} \)

2. **Define a Relation**:
- 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. **Find the Cartesian Product \( A \times B \)**:
- The Cartesian product \( A \times B \) is:
\[ A \times B = \{(x, a), (x, b), (x, c), (x, d), (y, a), (y, b), (y, c), (y, d), (z, a), (z, b), (z, c), (z, d)\} \]

4. **Analyze the Options**:
- We need to check each option provided to see if it is a subset of \( A \times B \). An option that contains an ordered pair not in \( A \times B \) is not a relation from A to B.

5. **Example Options**:
- Suppose the options are:
- A: \( \{(x, a), (x, c)\} \)
- B: \( \{(y, c), (y, d)\} \)
- C: \( \{(z, a), (z, d)\} \)
- D: \( \{(y, b), (z, e)\} \) (Assuming \( e \) is not in set B)

6. **Check Each Option**:
- Option A: \( \{(x, a), (x, c)\} \) is a relation (both pairs are in \( A \times B \)).
- Option B: \( \{(y, c), (y, d)\} \) is a relation (both pairs are in \( A \times B \)).
- Option C: \( \{(z, a), (z, d)\} \) is a relation (both pairs are in \( A \times B \)).
- Option D: \( \{(y, b), (z, e)\} \) is **not a relation** because \( (z, e) \) is not in \( A \times B \) (since \( e \) is not an element of set B).

7. **Conclusion**:
- Therefore, the option that is not a relation from A to B is **Option D**.