The void relation on a set $A$ is

On $R$, the relation $\rho$ is defined by '$x \rho y$ holds if and only if $x-y$ is zero or irrational'. Then,

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(x, y) \mid x, y \in R \text{ and } xy \text{ is an irrational number}\}$. Then,the relation $r$ is:

Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.

Define a relation $R$ on $A=\{1, 2, 3, 4\}$ as $x R y$ if $x$ divides $y$. $R$ is

Let $A$ be the non-void set of the children in a family. The relation '$x$ is a brother of $y$' on $A$ is