Let A=\ 2,3,6,8,9,11\ and B=\ 1,4,5,10,15\ Let R be a relation on A × B define by ( a , b ) R ( c , d ) if and only if 3 ad -7 bc is an even integer. Then the relation R is

reflexive and symmetric but not transitive.

Mathematics · Sets and Relations

JEE Main 2024 — 8 April, Shift 2 — Question 16

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

Option A:
reflexive but not symmetric.
Option B:
transitive but not symmetric.
Option C:
reflexive and symmetric but not transitive.
Correct
Option D:
an equivalence relation.

Answer: C

Step-by-step solution

$\quad A=\{2,3,6,8,9,11\} \quad(a, b) R(c, d)$

$B=\{1,4,5,10,15\} \quad 3 \mathrm{ad}-7 \mathrm{bc}$

Reflexive : $(a, b) R(a, b)$

$\Rightarrow 3 \mathrm{ab}-7 \mathrm{ba}=-4 \mathrm{ab}$

always even so it is reflexive.

Symmetric : If $3 \mathrm{ad}-7 \mathrm{bc}=$ Even

Case-I : odd odd

Case-II: even even

$(\mathrm{c}, \mathrm{d}) \mathrm{R}(\mathrm{a}, \mathrm{b}) \Rightarrow 3 \mathrm{bc}-3 \mathrm{ab}$

Case-I : odd odd

Case-II : even even

so symmetric relation Transitive :

Set $(3,4) R(6,4)$ Satisfy relation

Set $(6,4) R(3,1)$ Satisfy relation

but $(3,4) R(3,1)$ does not satisfy relation so not transitive.

Answer key and solution verified before publishing.

Practise Sets and Relations

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.

Solve a similar one free

Exam: JEE Main 2024
Paper: 8 April, Shift 2
Subject: Mathematics
Chapter: Sets and Relations
Topic: Types of Relations