The empty relation on a set $A$ is

Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) : \text{distance of the point } P \text{ from the origin is same as the distance of the point } Q \text{ from the origin}\}$,is an equivalence relation. Further,show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.

$A$ relation $R$ on the set of natural numbers is defined as $\{(a, b) : |a - b| = 3\}$. Then $R$ is:

The relation $S$ in the set $R$ of real numbers,defined as $S = \{(a, b) : a < b^2\}$ is a . . . . . . relation.

Let the relation $\rho$ be defined on $\mathbb{R}$ by $a \rho b$ if and only if $a-b$ is zero or irrational. Then:

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