Here, R₁, R₂, R_3, and R₄ are the binary relations.

So, recall that for any binary relation R on set A. We have,

R is reflexive if for all x ∈ A, xRx.

R is symmetric if for all x, y ∈ A, if xRy, then yRx.

R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.

So, using these results let us start determining given relations.

We have

R₁ on Q₀ defined by (a, b) ∈ R₁⇔

Check for Reflexivity:

∀ a, b ∈ Q₀,

(a, a), (b, b) ∈ R₁ needs to be proved for reflexivity.

If (a, b) ∈ R₁

Then, …(1)

So, for (a, a) ∈ R₁

Replace b by a in equation (1), we get

But, we know

⇒ (a, a) ∉ R₁

So, ∀ a ∈ Q₀, then (a, a) ∉ R₁

∴ R₁ is not reflexive.

Check for Symmetry:

If (a, b) ∈ R₁

Then, (b, a) ∈ R₁

∀ a, b ∈ Q₀

If (a, b) ∈ R₁

We have, …(2)

Now, for (b, a) ∈ R₁

Replace a by b & b by a in equation (2), we get

⇒ (b, a) ∈ R₂

So, if (a, b) ∈ R₁, then (b, a) ∈ R₁

∀ a, b ∈ Q₀

∴ R₁ is symmetric.

Check for Transitivity:

If (a, b) ∈ R₁ and (b, c) ∈ R₁

We need to eliminate b.

We have

Putting in , we get

But,

⇒ (a, c) ∉ R₁

So, if (a, b) ∈ R₁ and (b, c) ∈ R₁, then (a, c) ∉ R₁

∀ a, b, c ∈ Q₀

∴ R₁ is not transitive.