Question

Let A = {1, 2, 3}, and let R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R₂ = {(2, 2), (3, 1), (1, 3)}, R₃ = {(1, 3), (3, 3)}. Find whether or not each of the relations R₁, R₂, R₃ on A is

(i) reflexive

(ii) symmetric

(iii) transitive.

Answer 1

Consider as R₁

Given that R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}

Reflexivity:

Here, (1, 1), (2, 2), (3, 3) ∈ R

Clearly, R₁ is reflexive.

Symmetry:

Here, (2, 1) ∈ R₁,

But (1, 2) ∉ R₁

So, R₁ is not symmetric.

Transitivity:

Here, (2, 1) ∈ R₁ and (1, 3) ∈ R₁,

But (2, 3) ∉ R₁

So, R₁ is not transitive.

Now we consider R₂

Given that R₂ = {(2, 2), (3, 1), (1, 3)}

Reflexivity:

Clearly, (1, 1) and (3, 3) ∉ R₂

So, R₂ is not reflexive.

Symmetry:

Here, (1, 3) ∈ R₂ and (3, 1) ∈ R₂

Clearly, R₂ is symmetric.

Transitivity:

Here, (1, 3) ∈ R₂ and (3, 1) ∈ R₂

But (3, 3) ∉ R₂

So, R₂ is not transitive.

Consider as R₃

Given that R₃ = {(1, 3), (3, 3)}

Reflexivity:

Clearly, (1, 1) ∉ R₃

So, R₃ is not reflexive.

Symmetry:

Here, (1, 3) ∈ R₃, but (3, 1) ∉ R₃

So, R₃ is not symmetric.

Transitivity:

Here, (1, 3) ∈ R₃ and (3, 3) ∈ R₃

Also, (1, 3) ∈ R₃

Clearly, R₃ is transitive.