Show that relation R in R defined as R ={(a, b): a ≤ b}, is reflexive and transitive but not symme

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1.Relation and Function

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

Option A

Option B

Option C

Option D

Solution

Solution $4: R =\{( a , b ): a \leq b \}$

Clearly $(a, a) \in R$ $[$ as $a=a]$

$\therefore R$ is reflexive.

Now, $(2,4)\in R$ $($ as $2<4)$

But, $(4,2)\notin R$ as $4$ is greater than $2$.

$\therefore R$ is not symmetric. Now, let $(a, b),\,(b, c) \in R$

Then, $a \leq b$ and $b \leq c$

$\Rightarrow $ $a \leq c$

$\Rightarrow $ $(a, c) \in R$

$\therefore R$ is transitive.

Hence $R$ is reflexive and transitive but not symmetric

Standard 12

Mathematics

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