Find the domain and range of the relation `R={(x,x^(3)): x lt= 4, x in N}`

Answer

To find the domain and range of the relation \( R = \{(x, x^3) : x \leq 4, x \in \mathbb{N}\} \), we will follow these steps:

### Step 1: Identify the values of \( x \)
Since \( x \) belongs to the set of natural numbers \( \mathbb{N} \) and is less than or equal to 4, the possible values for \( x \) are:
- \( x = 1 \)
- \( x = 2 \)
- \( x = 3 \)
- \( x = 4 \)

### Step 2: Calculate \( x^3 \) for each value of \( x \)
Now, we will calculate \( x^3 \) for each of the values of \( x \):
- For \( x = 1 \): \( 1^3 = 1 \)
- For \( x = 2 \): \( 2^3 = 8 \)
- For \( x = 3 \): \( 3^3 = 27 \)
- For \( x = 4 \): \( 4^3 = 64 \)

### Step 3: Write the relation in roster form
The relation \( R \) can be expressed in roster form as:
\[ R = \{(1, 1), (2, 8), (3, 27), (4, 64)\} \]

### Step 4: Determine the domain
The domain of the relation consists of the first elements of each ordered pair in \( R \):
- Domain \( D = \{1, 2, 3, 4\} \)

### Step 5: Determine the range
The range of the relation consists of the second elements of each ordered pair in \( R \):
- Range \( R = \{1, 8, 27, 64\} \)

### Final Answer
- Domain: \( D = \{1, 2, 3, 4\} \)
- Range: \( R = \{1, 8, 27, 64\} \)

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