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, 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 \( A \times B \). The Cartesian product \( A \times B \) consists of all possible ordered pairs where the first element is from set \( A \) and the second element is from set \( B \).

**Calculation of \( A \times B \)**:
\[ A \times B = \{(1, 1), (1, 5), (1, 9), (1, 11), (1, 15), (1, 16), (2, 1), (2, 5), (2, 9), (2, 11), (2, 15), (2, 16), (3, 1), (3, 5), (3, 9), (3, 11), (3, 15), (3, 16), (4, 1), (4, 5), (4, 9), (4, 11), (4, 15), (4, 16)\} \]

Since \( f \) is a subset of \( A \times B \) (it contains pairs where the first element is from \( A \) and the second element is 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 \( B \) must satisfy the condition that every element in \( A \) is associated with exactly one element in \( B \). This means that for each \( a \in A \), there should be a unique \( b \in B \) such that \( (a, b) \in f \).

**Analysis of \( f \)**:
- From \( f \), we have:
- \( 1 \) maps to \( 5 \)
- \( 2 \) maps to both \( 9 \) and \( 11 \) (two distinct images)
- \( 3 \) maps to \( 1 \)
- \( 4 \) maps to \( 5 \)

Since the element \( 2 \) in set \( A \) is associated with two different elements in set \( B \) (i.e., \( 9 \) and \( 11 \)), it violates the definition 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 \( B \).