Let $\mathrm{A}=\{2,3,6,8,9,11\}$ and $\mathrm{B}=\{1,4,5,10,15\}$ 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

(C) reflexive and symmetric but not transitive.

(A) reflexive but not symmetric.

(B) transitive but not symmetric.

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Sets Relations and Functions

Let

A = {2, 3, 6, 8, 9, 11}

and

B = {1, 4, 5, 10, 15}

Let R be a relation on

A \times B

define by

(a, b) R (c, d)

if and only if

3 ad - 7 bc

is an even integer. Then the relation

R

Answer: (C)

Reflexive:

(a, b) R (a, b) \Leftrightarrow 3 ab - 7 ab = - 4 ab

, always even — reflexive.
Symmetric: if

3 a d - 7 b c

is even then checking parity cases shows

3 b c - 7 a d

(needed for

(c, d) R (a, b)

) is also even in each case — symmetric.
Transitive: testing

(3, 4) R (6, 4)

and

(6, 4) R (3, 1)

(both satisfy the relation) but

(3, 4) R (3, 1)

fails — not transitive.
So R is reflexive and symmetric but not transitive.

JEE Main · 2024 · 08 Apr 2024 · Shift 2 · Single correct option