Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by

$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$

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

Among the relations $S =\left\{( a , b ): a , b \in R -\{0\}, 2+\frac{ a }{ b } > 0\right\}$ And $T =\left\{( a , b ): a , b \in R , a ^2- b ^2 \in Z \right\}$,

Let $A=\{1,2,3\} .$ Then number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is

The relation $R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}$ is: