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On set $A=\{1,2,3\},$ relations $R$ and $S$ are given by $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$
$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$
Then,
Options:
$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$
Then,
Solution:
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Verified Answer
The correct answer is:
$R \cup S$ is reflexive and symmetric but not transitive
We have, $R=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$
$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$
$\therefore R \cup S=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}$
Since, $(2,1) \in R \cup S,(1,3) \in R \cup S$
but $(2,3) \in R \cup S$
$\therefore \mathrm{R} \cup \mathrm{S}$ is reflexive and symmetric but not transitive.
$S=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$
$\therefore R \cup S=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}$
Since, $(2,1) \in R \cup S,(1,3) \in R \cup S$
but $(2,3) \in R \cup S$
$\therefore \mathrm{R} \cup \mathrm{S}$ is reflexive and symmetric but not transitive.
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