JEE MAIN - Mathematics (2024 - 8th April Evening Shift - No. 13)
Let $$A=\{2,3,6,8,9,11\}$$ and $$B=\{1,4,5,10,15\}$$. Let $$R$$ be a relation on $$A \times B$$ defined by
$$(a, b) R(c, d)$$ if and only if $$3 a d-7 b c$$ is an even integer. Then the relation $$R$$ is
reflexive but not symmetric.
an equivalence relation.
reflexive and symmetric but not transitive.
transitive but not symmetric.
Explanation
$$(a, b) R(c, d) \Rightarrow 3 a d-7 b c \in$$ even
For reflexive
$$(a, b) R(a, b) \Rightarrow 3 a b-7 b a=-4 a b \in$$ even
For symmetric
$$(a, b) R(c, d)$$
_8th_April_Evening_Shift_en_13_1.png)
then
$$(c, d) R(a, b)=3 b c-7 a d$$
$$\in$$ even
Now check for transitive
$$\begin{aligned} & \begin{array}{ll} (a, b) R(c, d) \text { and }(c, d) R(e, f) \text { then }(a, b) R(e, f) \\ 3 a d-7 b c=2 m \quad \Rightarrow (2,5) R(6,8) \text { and } \\ 3 c f-7 e d=2 n \quad (6,8) R(9,4) \\ \text { then } 3 a f-7 e b \neq \text { even } \notin(2,5) R(9,4) \\ \Rightarrow \text { Not transitive option }(3) \end{array} \end{aligned}$$
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