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Show that the relation $\mathrm{R}$ in $\mathrm{R}$ defined as $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathbf{a} \leq \mathbf{b}\}$, is reflexive and transitive but not symmetric.
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$$
\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} \leq \mathrm{b}\}
$$
(i) $\mathrm{R}$ is reflexive, replacing b by $\mathrm{a}, \mathrm{a} \leq \mathrm{a} \Rightarrow \mathrm{a}=\mathrm{a}$ is true.
(ii) $\mathrm{R}$ is not symmetric, $\mathrm{a} \leq \mathrm{b}$, and $\mathrm{b} \leq \mathrm{a}$ which is not true $2 < 3$, but 3 is not less than 2 .
(iii) $\mathrm{R}$ is transitive, if $\mathrm{a} \leq \mathrm{b}$ and $\mathrm{b} \leq \mathrm{c}$, then $\mathrm{a} \leq \mathrm{c}$, e.g. $2 < 3,3 < 4 \Rightarrow 2 < 4$.
\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} \leq \mathrm{b}\}
$$
(i) $\mathrm{R}$ is reflexive, replacing b by $\mathrm{a}, \mathrm{a} \leq \mathrm{a} \Rightarrow \mathrm{a}=\mathrm{a}$ is true.
(ii) $\mathrm{R}$ is not symmetric, $\mathrm{a} \leq \mathrm{b}$, and $\mathrm{b} \leq \mathrm{a}$ which is not true $2 < 3$, but 3 is not less than 2 .
(iii) $\mathrm{R}$ is transitive, if $\mathrm{a} \leq \mathrm{b}$ and $\mathrm{b} \leq \mathrm{c}$, then $\mathrm{a} \leq \mathrm{c}$, e.g. $2 < 3,3 < 4 \Rightarrow 2 < 4$.
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