Relations and Functions — MATHS STD 12 Science — Question
Rajasthan 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
$R = \{(a,b) : a^3 \ge b^3\}$
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 $(a,b) (b,c) \in R$
$\Rightarrow a^{3 }\ge b^3$ and $b^{3 }\ge c^3$
$\Rightarrow a^{3 }\ge c^3$
$\Rightarrow (a,c) \in R$
$\Rightarrow R$ is transitive.
Therefore, relation $R$ is reflexive and transitive but not symmetric.
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