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

Answer

To determine whether the given set of ordered pairs \((z,a), (z,b), (z,c), (z,d)\) is a relation from set \(B\) to set \(A\), we will follow these steps:

### Step 1: Understand the Definition of a Relation
A relation from set \(B\) to set \(A\) is a subset of the Cartesian product \(B \times A\). This means each ordered pair in the relation must have its first element from set \(B\) and its second element from set \(A\).

### Step 2: Identify the Sets
We have:
- Set \(A = \{a, b, c, d\}\)
- Set \(B = \{x, y, z\}\)

### Step 3: Write the Ordered Pairs
The given relation is:
\[ \{(z,a), (z,b), (z,c), (z,d)\} \]

### Step 4: Check Each Ordered Pair
We will check whether each ordered pair in the relation has its first element in set \(B\) and its second element in set \(A\):
- For the pair \((z,a)\):
- First element \(z\) is in \(B\) (yes)
- Second element \(a\) is in \(A\) (yes)

- For the pair \((z,b)\):
- First element \(z\) is in \(B\) (yes)
- Second element \(b\) is in \(A\) (yes)

- For the pair \((z,c)\):
- First element \(z\) is in \(B\) (yes)
- Second element \(c\) is in \(A\) (yes)

- For the pair \((z,d)\):
- First element \(z\) is in \(B\) (yes)
- Second element \(d\) is in \(A\) (yes)

### Step 5: Conclusion
Since all ordered pairs have their first elements from set \(B\) and their second elements from set \(A\), we can conclude that the given set of ordered pairs is indeed a relation from \(B\) to \(A\).

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

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