Regional problem · Logic and sets

A binary relation \(\varrho\) on the set of real numbers is defined by

(\forall x, y \in \mathbb{R}) \quad x \varrho y \iff x^2 - 3xy + 2y^2 = 0 .

(a) Determine whether \(\varrho\) is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive.

(b) Decide whether \(\varrho\) is an equivalence relation on \(\mathbb{R}\). If it is, find the equivalence classes of the elements \(0\), \(1\), \(2\) and \(6\).

(c) Decide whether \(\varrho\) is an order relation on \(\mathbb{R}\), and whether it is a relation of total order on that set.