Let `R` be a relation defined on the set of natural numbers `N` as `R={(x ,\ y): x ,\ y in N ,\ 2x+y=41}` Find the domain and range of `R` . Also, verify whether `R` is (i) reflexive, (ii) symmetric (iii) transitive.

Explanation

To solve the problem, we need to find the domain and range of the relation \( R \) defined by \( R = \{(x, y) : x, y \in \mathbb{N}, 2x + y = 41\} \), and then verify whether \( R \) is reflexive, symmetric, and transitive.

### Step 1: Finding the Domain of \( R \)

The domain consists of all possible values of \( x \) such that there exists a corresponding \( y \) that satisfies the equation \( 2x + y = 41 \) and both \( x \) and \( y \) are natural numbers.

1. Rearranging the equation gives:
\[
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