Let `A={a,b,c,d}` and `B={x,y,z}`. Which of the following are relations from A to B? `{(a,x),(x,a),(b,y),(y,b)}`

Answer

To determine which of the given pairs are relations from set A to set B, we need to first understand what a relation from set A to set B is. A relation from A to 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 A and 'b' is an element from B.

### Step-by-Step Solution:

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

2. **Find the Cartesian Product \( A \times B \)**:
- The Cartesian product \( A \times B \) consists of all possible ordered pairs (a, b):
- \( A \times B = \{(a,x), (a,y), (a,z), (b,x), (b,y), (b,z), (c,x), (c,y), (c,z), (d,x), (d,y), (d,z)\} \)

3. **Given Relation**:
- The given relation is \( \{(a,x), (x,a), (b,y), (y,b)\} \).

4. **Check Each Pair**:
- **(a, x)**: This pair is valid as 'a' is in A and 'x' is in B.
- **(x, a)**: This pair is not valid as 'x' is not in A.
- **(b, y)**: This pair is valid as 'b' is in A and 'y' is in B.
- **(y, b)**: This pair is not valid as 'y' is not in A.

5. **Conclusion**:
- The valid pairs from the given relation that are from A to B are \( (a,x) \) and \( (b,y) \).
- Therefore, the pairs \( (x,a) \) and \( (y,b) \) are not relations from A to B.

### Final Answer:
The relations from A to B in the given set are \( \{(a,x), (b,y)\} \).