Determine the domain and range of the following relations : `{(x,y):x" is a multiple of 3 and y is a multiple of 5"}`

Answer

To determine the domain and range of the relation \(\{(x,y): x \text{ is a multiple of } 3 \text{ and } y \text{ is a multiple of } 5\}\), we can follow these steps:

### Step 1: Identify the domain
The domain consists of all possible values of \(x\) that satisfy the condition given in the relation. Here, \(x\) is defined as a multiple of 3.

- A multiple of 3 can be expressed as:
\[ x = 3n \]
where \(n\) is any integer (i.e., \(n \in \mathbb{Z}\)).

Thus, the domain can be written as:
\[ \text{Domain} = \{3n : n \in \mathbb{Z}\} \]

### Step 2: Identify the range
The range consists of all possible values of \(y\) that satisfy the condition given in the relation. Here, \(y\) is defined as a multiple of 5.

- A multiple of 5 can be expressed as:
\[ y = 5m \]
where \(m\) is any integer (i.e., \(m \in \mathbb{Z}\)).

Thus, the range can be written as:
\[ \text{Range} = \{5m : m \in \mathbb{Z}\} \]

### Final Result
Putting it all together, we have:
- **Domain**: \(\{3n : n \in \mathbb{Z}\}\)
- **Range**: \(\{5m : m \in \mathbb{Z}\}\)