Let a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right)…
Let a relation $\mathrm{R}$ on $\mathrm{N} \times N$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:
(I) $\mathrm{R}$ is reflexive but not symmetric.
(II) $R$ is transitive Then which one of the following is true?
Both (I) and (II) are correct.
Only (II) is correct.
Neither (I) nor (II) is correct.
Only (I) is correct.
Solution
All $\left(\left(\mathrm{x}_1 \mathrm{y}_1\right),\left(\mathrm{x}_1, \mathrm{y}_1\right)\right)$ are in $\mathrm{R}$ where
$\mathrm{x}_1, \mathrm{y}_1 \in \mathrm{N} \therefore \mathrm{R}$ is reflexive
$((1,1),(2,3)) \in \mathrm{R} \text { but }((2,3),(1,1)) \notin \mathrm{R}$
$\therefore \mathrm{R}$ is not symmetric
$((2,4),(3,3)) \in \mathrm{R}$ and $((3,3),(1,3)) \in \mathrm{R}$ but $((2,4)$,
$(1,3)) \notin \mathrm{R}$
$\therefore \mathrm{R}$ is not transitive