Which of the following relations are functions ?Give reason.
. `R={(1,1),(2,2),(3,3),(4,4),(4,5)}`

Answer

To determine whether the relation \( R = \{(1,1), (2,2), (3,3), (4,4), (4,5)\} \) is a function, we need to follow these steps:

### Step 1: Understand the Definition of a Function
A relation \( f: A \to B \) is a function if every element \( x \) in set \( A \) is associated with exactly one element \( y \) in set \( B \). This means that for each input, there must be a unique output.

### Step 2: Identify the Elements in the Relation
From the relation \( R \), we can extract the elements:
- The first elements (inputs) are: \( 1, 2, 3, 4, 4 \)
- The second elements (outputs) are: \( 1, 2, 3, 4, 5 \)

### Step 3: Check for Unique Outputs
Now, we need to check if each input has a unique output:
- For input \( 1 \): Output is \( 1 \)
- For input \( 2 \): Output is \( 2 \)
- For input \( 3 \): Output is \( 3 \)
- For input \( 4 \): There are two outputs, \( 4 \) and \( 5 \)

### Step 4: Conclusion
Since the input \( 4 \) corresponds to two different outputs \( 4 \) and \( 5 \), this violates the definition of a function, which requires each input to have exactly one output.

### Final Answer
The relation \( R \) is **not a function** because the element \( 4 \) in set \( A \) has two images \( 4 \) and \( 5 \) in set \( B \), which is not allowed for a function.

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