Question

If A = {1, 2, 3, 4 }, define relations on A which have properties of being:

(a) reflexive, transitive but not symmetric

(b) symmetric but neither reflexive nor transitive

(c) reflexive, symmetric and transitive.

Answer 1

Given that, A = {1, 2, 3}.

(i) Let R₁ = {(1, 1), (1, 2), (1, 3), (2, 3), (2, 2), (1, 3), (3, 3)}

R₁ is reflexive as (1, 1), (2, 2) and (3, 3) lie is R₁.

R₁ is transitive as (1, 2) ∈ R₁, (2, 3) ∈ R₁ ⇒ (1, 3) ∈ R₁

Now, (1, 2) ∈ R₁ ⇒ (2, 1) ∉ R₁.

(ii) Let R₂ = {(1, 2), (2, 1)}

Now, (1, 2) ∈ R₂, (2, 1) ∈ R₂

So, it is symmetric,

And, clearly R₂ is not reflexive as (1, 1) ∉ R₂

Also, R₂ is not transitive as (1, 2) ∈ R₂, (2, 1) ∈ R₂ but (1, 1) ∉ R₂

(iii) Let R₃ = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

R₃ is reflexive as (1, 1) (2, 2) and (3, 3) ∈ R₁

R₃ is symmetric as (1, 2), (1, 3), (2, 3) ∈ R₁ ⇒ (2, 1), (3, 1), (3, 2) ∈ R₁

Therefore, R₃ is reflexive, symmetric and transitive.