If $\mathrm{R}$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset R$, then the number of elements in $\mathrm{R}$ is

The minimum number of elements that must be added to the relation $R =\{( a , b ),( b , c )$, (b, d) $\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation, is $.........$

Given a non empty set $X$, consider $P ( X )$ which is the set of all subsets of $X$.

Define the relation $R$ in $P(X)$ as follows :

For subsets $A,\, B$ in $P(X),$ $ARB$ if and only if $A \subset B .$ Is $R$ an equivalence relation on $P ( X ) ?$ Justify your answer.

Let $N$ be the set of natural numbers and a relation $R$ on $N$ be defined by $R=\left\{(x, y) \in N \times N: x^{3}-3 x^{2} y-x y^{2}+3 y^{3}=0\right\} .$ Then the relation $R$ is:

Let $P ( S )$ denote the power set of $S =\{1,2,3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $\left( A \cap B ^{ c }\right) \cup\left( B \cap A ^{ c }\right)=\varnothing$ and $AR _2 B$ if $A \cup B ^{ c }=$ $B \cup A ^{ c }, \forall A , B \in P ( S )$. Then :

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.