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 step by step, we will first define the relation \( R \) and then find its domain and range. Finally, we will check if the relation is reflexive, symmetric, or transitive.

### Step 1: Define the Relation
The relation \( R \) is defined as:
\[ R = \{(x, y) : x, y \in \mathbb{N}, 2x + y = 41\} \]
where \( \mathbb{N} \) is the set of natural numbers.

### Step 2: Find the Domain
To find the domain, we need to determine the possible values of \( x \) such that \( y \) remains a natural number. Rearranging the equation gives:
\[ y = 41 - 2x \]
For \( y \) to be a natural number, \( 41 - 2x > 0 \). This implies:
\[ 41 > 2x \quad \Rightarrow \quad x < \frac{41}{2} = 20.5 \]
Since \( x \) must be a natural number, the maximum value for \( x \) is 20. Thus, \( x \) can take values from 1 to 20.

The domain of \( R \) is:
\[ \text{Domain} = \{1, 2, 3, \ldots, 20\} \]

### Step 3: Find the Range
Next, we will find the corresponding values of \( y \) for each \( x \) in the domain:
- For \( x = 1: y = 41 - 2(1) = 39 \)
- For \( x = 2: y = 41 - 2(2) = 37 \)
- For \( x = 3: y = 41 - 2(3) = 35 \)
- ...
- For \( x = 20: y = 41 - 2(20) = 1 \)

Thus, the values of \( y \) will be:
\[ \{39, 37, 35, \ldots, 1\} \]
This is an arithmetic sequence where the first term is 39 and the last term is 1, with a common difference of -2.

The range of \( R \) is:
\[ \text{Range} = \{1, 3, 5, \ldots, 39\} \]

### Step 4: Check for Reflexivity
A relation is reflexive if for every \( x \) in the domain, \( (x, x) \) is in \( R \).

We check:
\[ 2x + x = 41 \quad \Rightarrow \quad 3x = 41 \quad \Rightarrow \quad x = \frac{41}{3} \approx 13.67 \]
Since \( x \) is not a natural number, \( R \) is not reflexive.

### Step 5: Check for Symmetry
A relation is symmetric if for every \( (x, y) \in R \), \( (y, x) \) is also in \( R \).

For example, \( (1, 39) \in R \) but \( (39, 1) \notin R \). Thus, \( R \) is not symmetric.

### Step 6: Check for Transitivity
A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \).

For example, consider \( (20, 1) \in R \) and \( (1, 39) \in R \). However, \( (20, 39) \notin R \). Thus, \( R \) is not transitive.

### Final Conclusion
- **Domain**: \{1, 2, 3, ..., 20\}
- **Range**: \{1, 3, 5, ..., 39\}
- **Reflexive**: No
- **Symmetric**: No
- **Transitive**: No