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 $A=\{1,2,3\} .$ Then show that the number of relations containing $(1,2) $ and $(2,3)$ which are reflexive and transitive but not symmetric is four.

If $R$ is an equivalence relation on a set $A$, then ${R^{ - 1}}$ is

Let $A=\{1,2,3, \ldots 20\}$. Let $R_1$ and $R_2$ two relation on $\mathrm{A}$ such that $\mathrm{R}_1=\{(\mathrm{a}, \mathrm{b}): \mathrm{b}$ is divisible by $\mathrm{a}\}$ $\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}$ is an integral multiple of $\mathrm{b}\}$. Then, number of elements in $R_1-R_2$ is equal to_____.

Let $P = \{ (x,\,y)|{x^2} + {y^2} = 1,\,x,\,y \in R\} $. Then $P$ is

Show that the relation $R$ in $R$ defined as $R =\{(a, b): a \leq b\},$ is reflexive and transitive but not symmetric.