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

Let $\mathrm{T}$ be the set of all triangles in a plane with $\mathrm{R}$ a relation in $\mathrm{T}$ given by $\mathrm{R} =\left\{\left( \mathrm{T} _{1}, \mathrm{T} _{2}\right): \mathrm{T} _{1}\right.$ is congruent to $\left. \mathrm{T} _{2}\right\}$ . Show that $\mathrm{R}$ is an equivalence relation.

Let $A=\{1,2,3, \ldots \ldots .100\}$. Let $R$ be a relation on A defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $\mathrm{R} \subset \mathrm{R}_1$ and the number of elements in $\mathrm{R}_1$ is $\mathrm{n}$. Then, the minimum value of $n$ is..........................

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. 

Let L be the set of all lines in a plane and $\mathrm{R}$ be the relation in $\mathrm{L}$ defined as $\mathrm{R}=\left\{\left(\mathrm{L}_{1}, \mathrm{L}_{2}\right): \mathrm{L}_{1}\right.$ is perpendicular to $\left. \mathrm{L} _{2}\right\}$. Show that $\mathrm{R}$ is symmetric but neither reflexive nor transitive.

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