If A={a,b,c,d} and B={w,xy,z} , then which of the following is a relation from set A to B? Given reason:
`(i) R_(1)={(a,w),(a,x),(a,y),(a,z)}`
`(ii) R_(2)={(w,b),(z,c),(y,d)}`
(iii)`R_(3)={(a,y),(b,y),(w,y),(d,y)}`
(iv) `R_(4)={(b,z),(b,y),(c,z),(c,w)}`

Answer

To determine which of the given sets is a relation from set A to set B, we first need to understand what a relation is. 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 = {a, b, c, d}
- Set B = {w, x, y, z}

### Step 1: Find the Cartesian Product A × B

The Cartesian product A × B can be calculated as follows:
- Pair each element of A with each element of B:

- a with w, x, y, z → (a,w), (a,x), (a,y), (a,z)
- b with w, x, y, z → (b,w), (b,x), (b,y), (b,z)
- c with w, x, y, z → (c,w), (c,x), (c,y), (c,z)
- d with w, x, y, z → (d,w), (d,x), (d,y), (d,z)

Thus, the Cartesian product A × B is:
\[ A \times B = \{(a,w), (a,x), (a,y), (a,z), (b,w), (b,x), (b,y), (b,z), (c,w), (c,x), (c,y), (c,z), (d,w), (d,x), (d,y), (d,z)\} \]

### Step 2: Check Each Relation

Now we will check each of the given relations R1, R2, R3, and R4 to see if they are subsets of A × B.

#### (i) R1 = {(a,w), (a,x), (a,y), (a,z)}
- All pairs in R1 are present in A × B.
- Therefore, R1 is a relation from A to B.

#### (ii) R2 = {(w,b), (z,c), (y,d)}
- The pairs (w,b), (z,c), and (y,d) are not present in A × B because the first elements (w, z, y) are from set B, not A.
- Therefore, R2 is not a relation from A to B.

#### (iii) R3 = {(a,y), (b,y), (w,y), (d,y)}
- The pair (w,y) is not present in A × B because 'w' is from set B, not A.
- Therefore, R3 is not a relation from A to B.

#### (iv) R4 = {(b,z), (b,y), (c,z), (c,w)}
- The pairs (b,z), (b,y), (c,z), and (c,w) are not all present in A × B because 'z' and 'w' are from set B, and 'b' and 'c' are from set A.
- Therefore, R4 is not a relation from A to B.

### Conclusion

The relations from set A to set B are:
- R1 is a relation from A to B.
- R2, R3, and R4 are not relations from A to B.