Let $S$ be set of all real numbers ; then on set $S$ relation $R$ defined as $R = \{\ (a, b) : 1 + ab > 0\ \}$ is

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \, | a,b \in R$  and  $a - b + \sqrt 3$ is an irrational number$\}$ The relation $r$ is

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

In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation, it is sufficient, if $R$

Show that the relation $\mathrm{R}$ in the set $\mathrm{Z}$ of integers given by $\mathrm{R} =\{(\mathrm{a}, \mathrm{b}): 2$ divides $\mathrm{a}-\mathrm{b}\}$ is an equivalence relation.

If $A = \left\{ {x \in {z^ + }\,:x < 10} \right.$& and $x$ is a multiple of $3$ or $4\}$, where $z^+$ is the set of positive integers, then the total number of symmetric relations on $A$ is