R: x R y ⟺ is irrational.

Since, x-x=0 always, therefore is irrational. Hence, (x, x)∈R ∀ x ∈ R. Hence, R is reflexive.

R is not symmetric. Proof by counter-example:

Let . Then is irrational, Therefore (x, y)∈R. But

is rational, therefore (y, x)∉R. Hence, R is not symmetric.

R is not transitive. Proof by counter-example:

Let and , then

is irrational, therefore (x, y)∈R

is irrational, therefore (y, z)∈R

is rational, therefore (x, z)∉R. Hence, R is not transitive.