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

The number of reflexive relations of a set with four elements is equal to

The void relation on a set $A$ is

The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is

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 $A =\{1,2,3,4, \ldots .10\}$ and $B =\{0,1,2,3,4\}$ The number of elements in the relation $R =\{( a , b )$ $\left.\in A \times A : 2( a - b )^2+3( a - b ) \in B \right\}$ is $.........$.