Let A = {1,2,3,4}, B = { 1,5,9,11,15,16} and f = {(1,5), (2,9),(3,1),(4,5),(2,11)}. Then ,

Options

A. f is a relation from A to B
B. f is a function from A to B
C. Both (a) and (b)
D. None of these

Answer

To solve the problem, we need to analyze the sets and the relation given.

### Step 1: Identify the Sets and the Relation
We have:
- 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 2: Determine if \( f \) is a Relation from \( A \) to \( B \)
A relation from set \( A \) to set \( B \) means that every first element (from set \( A \)) in the ordered pairs of \( f \) must be an element of \( A \) and every second element (from set \( B \)) must be an element of \( B \).

- Check the first elements of the pairs in \( f \):
- \( 1 \) is in \( A \)
- \( 2 \) is in \( A \)
- \( 3 \) is in \( A \)
- \( 4 \) is in \( A \)
- \( 2 \) is in \( A \) again

Since all first elements are in \( A \), we check the second elements:
- Check the second elements of the pairs in \( f \):
- \( 5 \) is in \( B \)
- \( 9 \) is in \( B \)
- \( 1 \) is in \( B \)
- \( 5 \) is in \( B \) again
- \( 11 \) is in \( B \)

Since all second elements are in \( B \), we conclude that \( f \) is indeed a relation from \( A \) to \( B \).

### Step 3: Determine if \( f \) is a Function from \( A \) to \( B \)
For \( f \) to be a function from \( A \) to \( B \), each element in \( A \) must map to exactly one element in \( B \).

- Check the mappings:
- \( 1 \) maps to \( 5 \)
- \( 2 \) maps to \( 9 \) and \( 11 \) (two outputs)
- \( 3 \) maps to \( 1 \)
- \( 4 \) maps to \( 5 \)

Since the element \( 2 \) in \( A \) maps to two different elements in \( B \) (both \( 9 \) and \( 11 \)), \( f \) does not satisfy the condition of being a function.

### Conclusion
- \( f \) is a relation from \( A \) to \( B \).
- \( f \) is not a function from \( A \) to \( B \).

Thus, the correct option is that \( f \) is a relation from \( A \) to \( B \) but not a function.

### Final Answer
- \( f \) is a relation from \( A \) to \( B \) (Option A).