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JEE Main 20248 AprilEvening Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Sets and Relations

JEE Main 2024 · 8 April · Evening Shift · Mathematics · Sets and Relations

Let A=\ 2,3,6,8,9,11\ and B=\ 1,4,5,10,15\ . Let R be a relation on...

Let A={2,3,6,8,9,11} and B={1,4,5,10,15}A=\{2,3,6,8,9,11\} \text{ and }B=\{1,4,5,10,15\}. Let RR be a relation on A×BA \times B defined by (a,b)R(c,d)(a, b) R(c, d) if and only if 3ad−7bc3 a d-7 b c is an even integer. Then the relation RR is

Options

  1. Aan equivalence relation.
  2. Breflexive and symmetric but not transitive.
  3. Ctransitive but not symmetric.
  4. Dreflexive but not symmetric.

Correct answer

B. reflexive and symmetric but not transitive.

Step-by-step solution

A={2,3,6,8,9,11}(a,b)R(c,d)B={1,4,5,10,15}3ad−7bc\begin{array}{lc}\mathrm{A}=\{2,3,6,8,9,11\} & (\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d}) \\ \mathrm{B}=\{1,4,5,10,15\} & 3 \mathrm{ad}-7 \mathrm{bc}\end{array} Reflexive : (a,b)R(a,b)(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{a}, \mathrm{b}) ⇒3ab−7ba=−4ab\Rightarrow 3 \mathrm{ab}-7 \mathrm{ba}=-4 \mathrm{ab} always even so it is reflexive. Symmetric : If 3ad−7bc=3 \mathrm{ad}-7 \mathrm{bc}= Even Case-I : odd odd Case-II : even even (c, d) R(a,b)⇒3bc−3ab\mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow 3 \mathrm{bc}-3 \mathrm{ab} Case-I : odd odd Case-II : even even so symmetric relation Transitive : Set (3,4)R(6,4)(3,4) R(6,4) Satisfy relation Set (6,4)R(3,1)(6,4) R(3,1) Satisfy relation but (3,4)R(3,1)(3,4) \mathrm{R}(3,1) does not satisfy relation so not transitive.
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