Let $R$ and $S$ be two relations on a set $A$. Then

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

Let $A=\{0,3,4,6,7,8,9,10\} \quad$ and $R$ be the relation defined on A such that $R =\{( x , y ) \in A \times A : x - y \quad$ is odd positive integer or $x-y=2\}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to $...........$.

The relation $R$ defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(x, y)$ : $|{x^2} - {y^2}| < 16\} $ is given by

Let $P$ be the relation defined on the set of all real numbers such that 

$P = \left\{ {\left( {a,b} \right):{{\sec }^2}\,a - {{\tan }^2}\,b = 1\,} \right\}$. Then $P$ is

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is