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 A to B entails. A relation from set A to set B is a subset of the Cartesian product A × B, which consists of all possible ordered pairs (a, b) where 'a' is an element from set A and 'b' is an element from set B.

Given:
- Set A = {x, y, z}
- Set B = {a, b, c, d}

### Step-by-Step Solution:

1. **Identify the Cartesian Product A × B**:
The Cartesian product A × B includes all possible ordered pairs formed by taking each element from A and pairing it with each element from B.
\[ A \times B = \{(x,a), (x,b), (x,c), (x,d), (y,a), (y,b), (y,c), (y,d), (z,a), (z,b), (z,c), (z,d)\} \]

2. **List Possible Relations**:
A relation from A to B can be any subset of the Cartesian product A × B. For example, some possible relations could be:
- R1 = {(x,a), (y,b)}
- R2 = {(x,c), (z,d)}
- R3 = {(y,a), (y,c), (z,b)}

3. **Analyze the Given Options**:
We need to check the options provided (not given in the question) and see which one does not conform to the definition of a relation from A to B.

- If an option contains pairs where the first element is from B and the second from A, it is not a valid relation from A to B.
- For example, if an option contains (a,x) or (b,y), it represents a relation from B to A, not A to B.

4. **Identify the Incorrect Option**:
After analyzing the options, we find that one of them contains pairs like (a,x) or (b,y), which indicates that it is a relation from B to A. This option is not a valid relation from A to B.

5. **Conclusion**:
The option that contains pairs where the first element is from set B and the second from set A is the one that is not a relation from A to B.