Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

1.	Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Answer» Given as R = {(a, b): b = a + 1} Let us check for this relation whether it is reflexive, transitive and symmetric Reflexivity: Let a be an arbitrary element of R. Then, a = a + 1 cannot be true for all a ∈ A. ⇒ (a, a) ∉ R Therefore, R is not reflexive on A. Symmetry: Let (a, b) ∈ R ⇒ b = a + 1 ⇒ a = b − 1 Thus, (b, a) ∉ R Clearly, R is not symmetric on A. Transitivity: Let (1, 2) and (2, 3) ∈ R ⇒ 2 = 1 + 1 and 3 2 + 1 is true. But 3 ≠ 1+1 ⇒ (1, 3) ∉ R Therefore, R is not transitive on A.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

Answer»

Given as R = {(a, b): b = a + 1}

Let us check for this relation whether it is reflexive, transitive and symmetric

Reflexivity:

Let a be an arbitrary element of R.

Then, a = a + 1 cannot be true for all a ∈ A.

⇒ (a, a) ∉ R

Therefore, R is not reflexive on A.

Symmetry:

Let (a, b) ∈ R

⇒ b = a + 1

⇒ a = b − 1

Thus, (b, a) ∉ R

Clearly, R is not symmetric on A.

Transitivity:

Let (1, 2) and (2, 3) ∈ R

⇒ 2 = 1 + 1 and 3

2 + 1 is true.

But 3 ≠ 1+1 ⇒ (1, 3) ∉ R

Therefore, R is not transitive on A.