The relation $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \operatorname{gcd}(\mathrm{a}, \mathrm{b})=1,2 \mathrm{a} \neq \mathrm{b}, \mathrm{a}, \mathrm{b} \in \mathbb{Z}\}$ is:

(D) neither symmetric nor transitive

(A) transitive but not reflexive

(B) symmetric but not transitive

(C) reflexive but not symmetric

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Sets Relations and Functions

The relation

R = {(a, b) : gcd (a, b) = 1, 2 a \neq = b, a, b \in Z}

is:

Answer: (D)

For reflexivity

(a, a)

needs

g cd (a, a) = ∣ a ∣ = 1

, which fails for most integers, so

R

is not reflexive. For symmetry,

(2, 1) \in R

since

g cd (2, 1) = 1

and

2 \cdot 2 = 4 \neq = 1

, yet

(1, 2) \in / R

because

2 \cdot 1 = 2

equals the second entry, so

R

is not symmetric. For transitivity,

(14, 19)

and

(19, 21)

both have gcd

1

and satisfy

2 a \neq = b

, but

g cd (14, 21) = 7 \neq = 1

, so

(14, 21) \in / R

and

R

is not transitive. The relation is therefore neither symmetric nor transitive.

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option