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1.Relation and Function
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Let $R$ be a relation on $Z \times Z$ defined by$ (a, b)$$R(c, d)$ if and only if $ad - bc$ is divisible by $5$ . Then $\mathrm{R}$ is
A
Reflexive and symmetric but not transitive
B
Reflexive but neither symmetric not transitive
C
Reflexive, symmetric and transitive.
D
Reflexive and transitive but not symmetric
(JEE MAIN-2024)
Solution
$(a, b) R(a, b)$ as $a b-a b=0$
Therefore reflexive
Let $(a, b) R(c, d) \Rightarrow a d-b c$ is divisible by $5$
$\Rightarrow \mathrm{bc}-\mathrm{ad}$ is divisible by $5 \Rightarrow(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b})$
Therefore symmetric
Relation not transitive as $(3,1) \mathrm{R}(10,5)$ and $(10,5) \mathrm{R}(1,1)$ but $(3,1)$ is not related to $(1,1)$
Standard 12
Mathematics
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