Let $A =\{1,2,3,4, \ldots .10\}$ and $B =\{0,1,2,3,4\}$ The number of elements in the relation $R =\{( a , b )$ $\left.\in A \times A : 2( a – b )^2+3( a – b ) \in B \right\}$ is $………$.

Let ${R_1}$ be a relation defined by ${R_1} = \{ (a,\,b)|a \ge b,\,a,\,b \in R\} $. Then ${R_1}$ is

Let $A=\{1,2,3,4\}$ and $R$ be a relation on the set $A \times A$ defined by $R=\{((a, b),(c, d)): 2 a+3 b=4 c+5 d\}$. Then the number of elements in $R$ is:

Let $H$ be the set of all houses in a village where each house is faced in one of the directions, East, West, North, South. Let $R = \{ (x,y)|(x,y) \in H \times H$ and $x, y$ are faced in same direction $\}$ . Then the relation $' R '$ is

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