Let $N$ be the set of natural numbers greater than $100. $ Define the relation $R$ by : $R = \{(x,y) \in \,N × N :$ the numbers $x$ and $y$ have atleast two common divisors$\}.$ Then $R$ is-

Consider the relations $R_1$ and $R_2$ defined as $a R_1 b$ $\Leftrightarrow a^2+b^2=1$ for all $a, b, \in R$ and $(a, b) R_2(c, d)$ $\Leftrightarrow a+d=b+c$ for all $(a, b),(c, d) \in N \times N$. Then

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.

$R$ is a relation over the set of real numbers and it is given by $nm \ge 0$. Then $R$ is

The relation $R= \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)\}$ on set $A = \{1, 2, 3\}$ is