(i) Consider as R₁

Given that R₁ = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}

Now let us check R₁ is reflexive, symmetric and transitive

Reflexive:

Given (a, a), (b, b) and (c, c) ∈ R₁

Clearly, R₁ is reflexive.

Symmetric:

We see that the ordered pairs obtained by interchanging the components of R₁ are also in R₁.

So, R₁ is symmetric.

Transitive:

Here, (a, b) ∈ R₁, (b, c) ∈ R₁ and also (a, c) ∈ R₁

So, R₁ is transitive.

(ii) Consider as R₂

Given that R₂ = {(a, a)}

Reflexive:

Clearly (a, a) ∈ R₂.

So, R₂ is reflexive.

Symmetric:

Clearly (a, a) ∈ R ⇒ (a, a) ∈ R.

So, R₂ is symmetric.

Transitive:

R₂ is clearly a transitive relation, since there is only one element in it.

(iii) Consider as R₃

Given that R₃ = {(b, c)}

Reflexive:

Here,(b, b) ∉ R₃ neither (c, c) ∉ R₃

Clearly, R₃ is not reflexive.

Symmetric:

Here, (b, c) ∈ R₃, but (c, b) ∉ R₃

So, R₃ is not symmetric.

Transitive:

Here, R₃ has only two elements.

Hence, R₃ is transitive.

(iv) Consider as R₄

Given that R₄ = {(a, b), (b, c), (c, a)}.

Reflexive:

Here, (a, a) ∉ R₄, (b, b) ∉ R₄ (c, c) ∉ R₄

Clearly, R₄ is not reflexive.

Symmetric:

Here, (a, b) ∈ R₄, but (b, a) ∉ R₄.

So, R₄ is not symmetric

Transitive:

Here, (a, b) ∈ R₄, (b, c) ∈ R₄, but (a, c) ∉ R₄

Clearly, R₄ is not transitive.