Show that the relation R on defined as R= (a,b):a , is reflexive,… - Vidyadip

Question

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

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Answer

$\text{R}=\big\{(\text{a},\text{b}):\text{a}\leq\text{b}\big\}$
Clearly $(\text{a},\text{a})\in\text{R}$ as $\text{a}=\text{a}.$
$\therefore\ $R is reflexive.
Now,
$(2,4)\in\text{R}$ as $(2<4)$
But, $(4,2)\notin\text{R}$ R as 4 is greater than 2.
$\therefore\ $R is not symmetric.
Now, let $(\text{a},\text{b}),(\text{b},\text{c})\in\text{R}.$
Then,
$\text{a}\leq\text{b}$ and $\text{b}\leq\text{c}$
$\Rightarrow\text{a}\leq\text{c}$
$\Rightarrow(\text{a},\text{c})\in\text{R}$
$\therefore\ $R is transitive.
Hence, R is reflexive and transitive but not symmetric.

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