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

Answer

To determine whether the set of pairs \(\{(a,y),(a,z),(c,x),(d,y)\}\) represents a relation from set \(A\) to set \(B\), we need to analyze the definitions of the sets and the pairs.

### Step-by-Step Solution:

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

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 \(A\) and the second element is from \(B\).

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

4. **List the Given Relation**:
- The given relation is \(\{(a,y), (a,z), (c,x), (d,y)\}\).

5. **Check Each Pair**:
- Check if each pair in the given relation is an element of \(A \times B\):
- \((a,y)\): \(a \in A\) and \(y \in B\) → valid
- \((a,z)\): \(a \in A\) and \(z \in B\) → valid
- \((c,x)\): \(c \in A\) and \(x \in B\) → valid
- \((d,y)\): \(d \in A\) and \(y \in B\) → valid

6. **Conclusion**:
- Since all pairs in the given relation are valid pairs from \(A\) to \(B\), we conclude that \(\{(a,y), (a,z), (c,x), (d,y)\}\) is indeed a relation from \(A\) to \(B\).

### Final Answer:
The set \(\{(a,y), (a,z), (c,x), (d,y)\}\) is a relation from \(A\) to \(B\).