Let R be relation defined on the set of natural number N as follows: R = { ( x , y ) : x ∈ N , y ∈ N , 2 x + y = 41 } . Find the domain and range of the relation R . Also verify whether R is reflexive, symmetric and transitive.

Given that, R = { ( x , y ) : x ∈ N , y ∈ N , 2 x + y = 41 } ⇒ R = 1 , 39 , 2 , 37 , 3 , 35 , . . . . 19 , 3 , 20 , 1 Domain = { 1 , 2 , 3 … . 20 } Range = { 1 , 3 , 5 , 7 , … … . 39 } Since, 2 × 2 + 2 ≠ 41 So, R is not reflexive as ( 2 , 2 ) ∉ R . As ( 1 , 39 ) ∈ R but ( 39 , 1 ) ∉ R So, R is not symmetric. Also, as ( 11 , 19 ) ∈ R , ( 19 , 3 ) ∈ R But ( 11 , 3 ) ∉ R . So, R is not transitive Hence, R is neither reflexive, nor symmetric and nor transitive.

Let R be relation defined on the set of natural number N as follows: R = { ( x , y ) : x ∈ N , y ∈ N , 2 x + y = 41 } . Find the domain and range of the relation R . Also verify whether R is reflexive, symmetric and transitive.

Given that, R = { ( x , y ) : x ∈ N , y ∈ N , 2 x + y = 41 } ⇒ R = 1 , 39 , 2 , 37 , 3 , 35 , . . . . 19 , 3 , 20 , 1 Domain = { 1 , 2 , 3 … . 20 } Range = { 1 , 3 , 5 , 7 , … … . 39 } Since, 2 × 2 + 2 ≠ 41 So, R is not reflexive as ( 2 , 2 ) ∉ R . As ( 1 , 39 ) ∈ R but ( 39 , 1 ) ∉ R So, R is not symmetric. Also, as ( 11 , 19 ) ∈ R , ( 19 , 3 ) ∈ R But ( 11 , 3 ) ∉ R . So, R is not transitive Hence, R is neither reflexive, nor symmetric and nor transitive.