A relation f on the set N of natural numbers is defined by `f={(n,n+3):ninN}`
The range of f = N - ….

Answer

To find the range of the relation \( f \) defined on the set of natural numbers \( \mathbb{N} \) by \( f = \{(n, n+3) : n \in \mathbb{N}\} \), we can follow these steps:

### Step 1: Understand the Relation
The relation \( f \) consists of ordered pairs where the first element \( n \) is a natural number and the second element is \( n + 3 \). This means for every natural number \( n \), there is a corresponding output \( n + 3 \).

### Step 2: Identify the Domain
The domain of \( f \) is the set of all natural numbers \( \mathbb{N} \). This includes numbers like 1, 2, 3, and so on.

### Step 3: Calculate the Range
To find the range, we compute the output for various values of \( n \):
- If \( n = 1 \), then \( f(1) = 1 + 3 = 4 \)
- If \( n = 2 \), then \( f(2) = 2 + 3 = 5 \)
- If \( n = 3 \), then \( f(3) = 3 + 3 = 6 \)
- If \( n = 4 \), then \( f(4) = 4 + 3 = 7 \)
- Continuing this way, we see that for any natural number \( n \), the output \( n + 3 \) will always be a natural number starting from 4.

### Step 4: Determine the Range
The outputs start from 4 and go on indefinitely. Therefore, the range of \( f \) can be expressed as:
\[ \text{Range of } f = \{ n \in \mathbb{N} : n \geq 4 \} \]
This can also be written as:
\[ \text{Range of } f = \mathbb{N} - \{1, 2, 3\} \]

### Final Answer
Thus, the range of the relation \( f \) is:
\[ \text{Range of } f = \mathbb{N} - \{1, 2, 3\} \]

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