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.

Explanation

To determine whether the statements regarding the relation \( f \) from set \( A \) to set \( B \) are true, 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 \( B \) is defined as a subset of the Cartesian product \( A \times B \). This means that every ordered pair in \( f \) must have its first element from set \( A \) and its second element from set \( B \).

- Ordered 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 ordered pairs in \( f \) have their first elements from \( A \) and second elements from \( B \), we conclude that:

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

### 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 \( A \) is associated with exactly one element in \( B \). This means that no element in \( A \) can map to more than one element in \( B \).

- Elements in \( A \):
- 1 maps to 5 (1 → 5) ✔️
- 2 maps to 9 and 11 (2 → 9 and 2 → 11) ❌
- 3 maps to 1 (3 → 1) ✔️
- 4 maps to 5 (4 → 5) ✔️

Here, the element 2 from set \( A \) maps to two different elements in set \( B \) (9 and 11). Therefore, it does not satisfy the condition of a function.

**Conclusion for Statement (ii):**
No, \( f \) is not a function from \( A \) to \( B \).

### Final Answers:
(i) True, \( f \) is a relation from \( A \) to \( B \).
(ii) False, \( f \) is not a function from \( A \) to \( B \).

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