If A={1,2,3,4},B={1,5,9,11,15,16} and `f={(1,5),(2,9),(3,1),(4,5),(2,11)}` are the following statements true ?
(i) f is a relation from A to B .
(ii) f is a function from A to B
Justify your answer.

Answer

To determine whether the statements regarding the relation \( f \) from set \( A \) to set \( B \) are true or false, we will analyze each statement step by step.

### Given:
- Set \( A = \{1, 2, 3, 4\} \)
- Set \( B = \{1, 5, 9, 11, 15, 16\} \)
- Relation \( f = \{(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)\} \)

### Step 1: Check if \( f \) is a relation from \( A \) to \( B \)

A relation from set \( A \) to set \( B \) consists of ordered pairs where the first element is from \( A \) and the second element is from \( B \).

- **Pairs in \( f \)**:
- \( (1, 5) \): 1 is in \( A \) and 5 is in \( B \) ✔️
- \( (2, 9) \): 2 is in \( A \) and 9 is in \( B \) ✔️
- \( (3, 1) \): 3 is in \( A \) and 1 is in \( B \) ✔️
- \( (4, 5) \): 4 is in \( A \) and 5 is in \( B \) ✔️
- \( (2, 11) \): 2 is in \( A \) and 11 is in \( B \) ✔️

Since all pairs in \( f \) have their first elements in \( A \) and second elements in \( B \), we conclude that:

**Conclusion for Statement (i)**: \( f \) is a relation from \( A \) to \( B \).
**Answer**: True.

### Step 2: Check if \( f \) is a function from \( A \) to \( B \)

A function from set \( A \) to set \( B \) requires that every element in \( A \) has a unique image in \( B \). This means that no element in \( A \) can be associated with more than one element in \( B \).

- **Examine the images**:
- For \( 1 \): Image is \( 5 \) (unique)
- For \( 2 \): Images are \( 9 \) and \( 11 \) (not unique)
- For \( 3 \): Image is \( 1 \) (unique)
- For \( 4 \): Image is \( 5 \) (unique)

Since the element \( 2 \) in set \( A \) has two images \( 9 \) and \( 11 \) in set \( B \), it does not satisfy the condition of having a unique image.

**Conclusion for Statement (ii)**: \( f \) is not a function from \( A \) to \( B \).
**Answer**: False.

### Final Answers:
(i) True
(ii) False