Let R be a relation on $\mathrm{Z} \times \mathrm{Z}$ defined by $(\mathrm{a}, \mathrm{b}) \mathrm{R}(\mathrm{c}, \mathrm{d})$ if and only if $\mathrm{ad}-\mathrm{bc}$ is divisible by 5. Then R is

(B) Reflexive but neither symmetric not transitive

(C) Reflexive, symmetric and transitive

(D) Reflexive and transitive but not symmetric

JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Sets Relations and Functions

Let R be a relation on

Z \times Z

defined by

(a, b) R (c, d)

if and only if

ad - bc

is divisible by 5.
Then R is

Answer: (A)

(a, b) R (c, d) ⟺ 5 ∣ (a d - b c)

.

Reflexive:

(a, b) R (a, b)

since

ab - ba = 0

, divisible by 5.

Symmetric: if

5 ∣ (a d - b c)

, then

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

is also divisible by 5, so

(c, d) R (a, b)

.

Transitive: fails. Take

(3, 1) R (10, 5)

3 (5) - 1 (10) = 5

, divisible by 5. And

(10, 5) R (1, 1)

10 (1) - 5 (1) = 5

, divisible by 5. But

(3, 1) R (1, 1)

3 (1) - 1 (1) = 2

, not divisible by 5. So transitivity fails.

Key confirmed: A (reflexive and symmetric but not transitive).

JEE Main · 2024 · 29 Jan 2024 · Shift 1 · Single correct option