If `A={1,2,3} and B={4,5,6}`, then which of the following is a relation from set A to B? Give reason:
(i) `R_(1)={(1,5),(2,4),(3,5)}`
(ii) `R_(2)={(4,1),(2,6),(5,1),(2,4)}`
(iii) `R_(3)={(1,4),(2,5),(3,4),(2,6),(3,6)}`
(iv) `AxxB`

Explanation

To determine which of the given sets is a relation from set A to set B, we need to understand that a relation from set A to set B is defined as a set of ordered pairs (x, y) where x belongs to set A and y belongs to set B.

Given:
- Set A = {1, 2, 3}
- Set B = {4, 5, 6}

Now, let's analyze each relation:

### (i) R₁ = {(1, 5), (2, 4), (3, 5)}
1. Check the first elements of the pairs:
- (1, 5): 1 ∈ A
- (2, 4): 2 ∈ A
- (3, 5): 3 ∈ A
2. Check the second elements of the pairs:
- (1, 5): 5 ∈ B
- (2, 4): 4 ∈ B
- (3, 5): 5 ∈ B
3. Since all first elements are from A and all second elements are from B, R₁ is a relation from A to B.

### (ii) R₂ = {(4, 1), (2, 6), (5, 1), (2, 4)}
1. Check the first elements of the pairs:
- (4, 1): 4 ∉ A
- (2, 6): 2 ∈ A
- (5, 1): 5 ∉ A
- (2, 4): 2 ∈ A
2. Since the first elements 4 and 5 do not belong to A, R₂ is not a relation from A to B.

### (iii) R₃ = {(1, 4), (2, 5), (3, 4), (2, 6), (3, 6)}
1. Check the first elements of the pairs:
- (1, 4): 1 ∈ A
- (2, 5): 2 ∈ A
- (3, 4): 3 ∈ A
- (2, 6): 2 ∈ A
- (3, 6): 3 ∈ A
2. Check the second elements of the pairs:
- (1, 4): 4 ∈ B
- (2, 5): 5 ∈ B
- (3, 4): 4 ∈ B
- (2, 6): 6 ∉ B
- (3, 6): 6 ∉ B
3. Since the second elements 6 do not belong to B, R₃ is not a relation from A to B.

### (iv) A × B
1. The Cartesian product A × B is defined as the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
2. Therefore, A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}.
3. Since all pairs in A × B have their first elements from A and second elements from B, A × B is a relation from A to B.

### Conclusion:
- R₁ is a relation from A to B.
- R₂ is not a relation from A to B.
- R₃ is not a relation from A to B.
- A × B is a relation from A to B.