Let R be a relation on the set of natural numbers N, defined as:
`R={(x,y):y=2x,x,y in N}.`
Is R a function from `N xxN`? If yes find the domain, co-domain and range of R.

Explanation

To determine whether the relation \( R \) defined as \( R = \{(x, y) : y = 2x, x, y \in \mathbb{N}\} \) is a function from \( \mathbb{N} \times \mathbb{N} \), we will follow these steps:

### Step 1: Check if \( R \) is a function
A relation \( R \) is a function if every element in the domain maps to exactly one element in the codomain.

Given the relation \( R \):
- For each \( x \in \mathbb{N} \), there exists a unique \( y \) such that \( y = 2x \).
- For example:
- If \( x = 1 \), then \( y = 2 \times 1 = 2 \).
- If \( x = 2 \), then \( y = 2 \times 2 = 4 \).
- If \( x = 3 \), then \( y = 2 \times 3 = 6 \).

Since each natural number \( x \) has a unique corresponding \( y \), the relation \( R \) is indeed a function.

### Step 2: Identify the domain of \( R \)
The domain of a function is the set of all possible inputs (or \( x \) values) that can be used in the function.

- Here, since \( x \) can be any natural number, the domain is:
\[ \text{Domain} = \mathbb{N} = \{1, 2, 3, 4, 5, \ldots\} \]

### Step 3: Identify the codomain of \( R \)
The codomain of a function is the set of all possible outputs (or \( y \) values) that the function can produce.

- Since \( y = 2x \) and \( x \) is a natural number, \( y \) will also be a natural number. Thus, the codomain is:
\[ \text{Codomain} = \mathbb{N} = \{1, 2, 3, 4, 5, \ldots\} \]

### Step 4: Identify the range of \( R \)
The range of a function is the set of all actual outputs produced by the function for the given domain.

- As \( x \) takes values from \( \mathbb{N} \), \( y \) takes values \( 2, 4, 6, 8, \ldots \), which are all the even natural numbers. Therefore, the range is:
\[ \text{Range} = \{y \in \mathbb{N} : y \text{ is even}\} = \{2, 4, 6, 8, \ldots\} \]
In set-builder notation, this can be expressed as:
\[ \text{Range} = \{2x : x \in \mathbb{N}\} \]

### Final Summary
- **Is \( R \) a function?** Yes.
- **Domain:** \( \mathbb{N} \)
- **Codomain:** \( \mathbb{N} \)
- **Range:** \( \{2x : x \in \mathbb{N}\} \) (even natural numbers)