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

Let $X$ be a family of sets and $R$ be a relation on $X$ defined by '$A$ is disjoint from $B$'. Then $R$ is

Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) : \text{distance of the point } P \text{ from the origin is same as the distance of the point } Q \text{ from the origin}\}$,is an equivalence relation. Further,show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.

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 $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and $R$ be a relation on $A$ defined by $x R y$ if and only if $2x - y \in \{0, 1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in $R$ to make it reflexive and symmetric relations,respectively. Then $l + m + n$ is equal to :-

Let $r$ be a relation from $R$ (set of real numbers) to $R$ defined by $r = \{(a,b) \mid a,b \in R \text{ and } a - b + \sqrt{3} \text{ is an irrational number} \}$. The relation $r$ is