Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$. Let $R$ be a relation on $A$ defined by $x R y$ if and only if $0 \leq x^2 + 2y \leq 4$. Let $l$ be the number of elements in $R$ and $m$ be the minimum number of elements required to be added to $R$ to make it a reflexive relation. Then $l+m$ is equal to

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

Let $L$ be the set of all straight lines in the Euclidean plane. Two lines $l_1$ and $l_2$ are related by the relation $R$ if and only if $l_1$ is parallel to $l_2$. Then the relation $R$ is:

$R = \{(\pi, \pi), (\pi^2, \pi^2), (\pi^3, \pi^3), (\pi, \pi^2), (\pi^2, \pi^3)\}$ is defined on the set $A = \{\pi, \pi^2, \pi^3\}$. 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.

Let $R$ and $S$ be two equivalence relations on a non-void set $A$. Then