Let R be a relation on \(\mathbb{N} \times \mathbb{N}\) defined by \((a, b) R (c, d)\) if and only if \(ad(b-c)=bc(a-d)\). Then R is

(B) transitive but neither reflexive nor symmetric

(C) reflexive and symmetric but not transitive

(D) symmetric and transitive but not reflexive

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Sets Relations and Functions

Let R be a relation on

N \times N

defined by

(a, b) R (c, d)

if and only if

a d (b - c) = b c (a - d)

. Then R is

Answer: (A)

Reflexivity needs

(a, b) R (a, b)

, i.e.

ab (b - a) = ab (a - b)

, which forces

a = b

and so fails in general. For symmetry, the defining relation

a d (b - c) = b c (a - d)

expands to

ab d - a c d - ab c + b c d = 0

, a form unchanged when

(a, b)

and

(c, d)

are interchanged, so the relation is symmetric. It is not transitive:

(2, 3) R (3, 2)

and

(3, 2) R (5, 30)

both hold, yet for

(2, 3)

and

(5, 30)

one side gives

2 \cdot 30 (3 - 5) = - 120

while the other gives

3 \cdot 5 (2 - 30) = - 420

, so that pair is unrelated. Thus the relation is symmetric but neither reflexive nor transitive.

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Single correct option