Give an example of a relation. Which is Symmetric and transitive but not reflexive.

Let $R = \{ (3,\,3),\;(6,\;6),\;(9,\,9),\;(12,\,12),\;(6,\,12),\;(3,\,9),(3,\,12),\,(3,\,6)\} $ be a relation on the set $A = \{ 3,\,6,\,9,\,12\} $. The relation is

If $R$ is a relation from a set $A$ to a set $B$ and $S$ is a relation from $B$ to a set $C$, then the relation $SoR$

Let $\mathrm{A}=\{1,2,3,4,5\}$. Let $\mathrm{R}$ be a relation on $\mathrm{A}$ defined by $x R y$ if and only if $4 x \leq 5 y$. Let $m$ be the number of elements in $\mathrm{R}$ and $\mathrm{n}$ be the minimum number of elements from $\mathrm{A} \times \mathrm{A}$ that are required to be added to $\mathrm{R}$ to make it a symmetric relation. Then $m+n$ is equal to:

Let $\mathrm{A}$ be the set of all students of a boys school. Show that the relation $\mathrm{R}$ in A given by $\mathrm{R} =\{(a, b): \mathrm{a} $ is sister of $\mathrm{b}\}$ is the empty relation and $\mathrm{R} ^{\prime}=\{(a, b)$ $:$ the difference between heights of $\mathrm{a}$ and $\mathrm{b}$ is less than $3\,\mathrm{meters}$ $\}$ is the universal relation. 

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