If A = { ( 0 , 1 , 2 , … , 9 } and the relation R on A is defined by R = { ( x , y ) : x , y ∈ A , y = 2 x + 1 } , then determine whether the relation R is reflexive.

Given that, A = { 0 , 1 , 2 , … , 9 } and relation R = { ( x , y ) ; x , y ∈ A , y = 2 x + 1 } . We know that, The reflexive property states that any real number a , is equal to itself. That is, a = a . Reflexive: Here, 1 ≠ 2 × 1 + 1 = 3 ∴ ( 1 , 1 ) ∉ R ∴ R is not reflexive on A .

If A = { ( 0 , 1 , 2 , … , 9 } and the relation R on A is defined by R = { ( x , y ) : x , y ∈ A , y = 2 x + 1 } , then determine whether the relation R is symmetric.

Given that, A = { 0 , 1 , 2 , … , 9 } and relation R = { ( x , y ) ; x , y ∈ A , y = 2 x + 1 } . We know that, the symmetric property states that for any real numbers, a and b , if a = b then b = a Symmetric: Here, ( 1 , 3 ) ∈ R Since, 3 = 2 · 1 + 1 But, ( 3 - 1 ) ∉ R ∵ 1 ≠ 2 × 3 + 1 = 7 ∴ R is not symmetric.

If A = { ( 0 , 1 , 2 , … , 9 } and the relation R on A is defined by R = { ( x , y ) : x , y ∈ A , y = 2 x + 1 } , then determine whether the relation R is transitive.

Given that, A = { 0 , 1 , 2 , … , 9 } and relation R = { ( x , y ) ; x , y ∈ A , y = 2 x + 1 } . Transitive: Here, ( 1 , 3 ) ∈ R ∵ 3 = 2 · 1 + 1 = 3 And ( 3 , 7 ) ∈ R ∵ 7 = 2 × 3 + 1 = 7 But ( 1 , 7 ) ∉ R ∵ 7 ≠ 2 × 1 + 1 = 3 ∴ For any 1 , 3 , 7 ∈ A , ( 1 , 3 ) , ( 3 , 7 ) ∈ R But ( 1 , 7 ) ∉ R Thus R is not transitive on A .