Relations and Functions — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsRelations and Functions2 Marks
Question
Give an example of a relation which is reflexive and transitive but not symmetric.
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Answer
Let us define a relation $R$ in $R$ as
$R=\left\{(a, b): a^3 \geq b^3\right\}$
It is clear that $(a, a) \in R$ as $a^3=a^3$
$\Rightarrow R$ is reflexive.
Now, $(2,1) \in R$, but $(1,2) \notin R$
$\Rightarrow R$ is not symmetric.
Now, let $(a, b)(b, c) \in R$
$ \Rightarrow a^3 \geq b^3 \text { and } b^3 \geq c^3 $
$ \Rightarrow a^3 \geq c^3 $
$ \Rightarrow(a, c) \in R $
$ \Rightarrow R \text { is transitive. }$
Therefore, relation R is reflexive and transitive but not symmetric.
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