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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?
Only ($II$) is correct.
Only ($I$) is correct.
Both ($I$) and ($II$) are correct.
Neither ($I$) nor ($II$) 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
$((1,1),(2,3)) \in \mathrm{R}$ 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
