$R$ is a relation from $\{11, 12, 13\}$ to $\{8, 10, 12\}$ defined by $y = x – 3$. Then ${R^{ – 1}}$ is

Let $r$ be a relation from $R$ (Set of real number) to $R$ defined by $r$ = $\left\{ {\left( {x,y} \right)\,|\,x,\,y\, \in \,R} \right.$ and $xy$ is an irrational number $\}$ , then relation $r$ is

The probability that a relation $R$ from $\{ x , y \}$ to $\{ x , y \}$ is both symmetric and transitive, is equal to

Let a relation $R$ on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)$ if and only if $\mathrm{x}_1 \leq \mathrm{x}_2$ or $\mathrm{y}_1 \leq \mathrm{y}_2$

Consider the two statements :

($I$) $\mathrm{R}$ is reflexive but not symmetric.

($II$) $\mathrm{R}$ is transitive

Then which one of the following is true?

Which of the following is not correct for relation $\mathrm{R}$ on the set of real numbers ?