Give an example of a relation. Which is Reflexive and transitive but not symmetric.

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1.Relation and Function

easy

Option A

Option B

Option C

Option D

Solution

Define a relation $R$ in $R$ as $:$

$\left.R=\{a, b): a^{3} \geq b^{3}\right\}$

Clearly $(a,a)\in R$ as $a^{3}=a^{3}$

$\therefore R$ is reflexive.

Now, $(2,1)\in R$ $[$ as $2^{3} \geq 1^{3}]$

But, $(1,2)\notin R$ $[$ as $1^{3} < 2^{3}]$

$\therefore R$ is not symmetric.

Now, Let $(a, b),\,(b, c) \in R$

$\Rightarrow a^{3} \geq b^{3}$ and $b^{3} \geq c^{3}$

$\Rightarrow a^{3} \geq c^{3}$

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

$\therefore R$ is transitive.

Hence, relation $R$ is reflexive and transitive but not symmetric.

Standard 12

Mathematics

medium

medium

(JEE MAIN-2025)

normal

medium

medium