Define a relation R on the set N of natural numbers by `R" "=" "{(x ," "y)" ":" "y" "=" "x" "+" "5` , x is a natural number less than 4; x, `y in N` }. Depict this relationship using roster form. Write down the domain and the range.

Explanation

To define the relation \( R \) on the set of natural numbers \( N \) as given in the question, we will follow these steps:

### Step 1: Identify the values of \( x \)
The relation is defined as \( R = \{(x, y) : y = x + 5, x \text{ is a natural number less than } 4\} \).
Since \( x \) is a natural number less than 4, the possible values for \( x \) are:
- \( x = 1 \)
- \( x = 2 \)
- \( x = 3 \)

### Step 2: Calculate corresponding values of \( y \)
Now, we will calculate \( y \) for each value of \( x \) using the relation \( y = x + 5 \):
- For \( x = 1 \):
\[ y = 1 + 5 = 6 \]
- For \( x = 2 \):
\[ y = 2 + 5 = 7 \]
- For \( x = 3 \):
\[ y = 3 + 5 = 8 \]

### Step 3: Write the relation in roster form
Now we can write the relation \( R \) in roster form:
\[ R = \{(1, 6), (2, 7), (3, 8)\} \]

### Step 4: Determine the domain
The domain of the relation \( R \) consists of all the first elements of the ordered pairs:
\[ \text{Domain} = \{1, 2, 3\} \]

### Step 5: Determine the range
The range of the relation \( R \) consists of all the second elements of the ordered pairs:
\[ \text{Range} = \{6, 7, 8\} \]

### Final Result
Thus, the relation \( R \) in roster form is:
\[ R = \{(1, 6), (2, 7), (3, 8)\} \]
The domain is:
\[ \text{Domain} = \{1, 2, 3\} \]
And the range is:
\[ \text{Range} = \{6, 7, 8\} \]