On R , the set of real numbers, a relation is defined as a b if and only if 1+a b>0 . Then,

Correct answer: C. is reflexive and symmetric but not transitive We know that, 1+a^ 2 >0, a ∈ R (a, a) ∈ is reflexive Again, let (a, b) ∈ 1+a b>0 1+b a>0 (b, a) ∈ is symmetric Now, (1,-0 · 1) ∈ and (-0 · 1,-9) ∈ but (1,-9) is not transitive.

JEE Advanced

Exam: JEE Advanced
Subject: Mathematics
Chapter: Sets and Relations

JEE Advanced · Mathematics · Sets and Relations

On R , the set of real numbers, a relation is defined as a b if and...

R

, the set of real numbers, a relation

\rho

is defined as

a \rho b

if and only if

1+a b>0

. Then,

Options

A $\rho$ is an equivalence relation
B $\rho$ is reflexive and transitive but not symmetric
C $\rho$ is reflexive and symmetric but not transitive
D $\rho$ is only symmetric

Correct answer

C. $\rho$ is reflexive and symmetric but not transitive

Step-by-step solution

We know that,

1+a^{2}>0, a \in R

\Rightarrow \quad(a, a) \in \rho

\therefore \rho

is reflexive
Again,

\quad

let

(a, b) \in \rho

\Rightarrow \quad 1+a b>0

\Rightarrow \quad 1+b a>0

\Rightarrow \quad(b, a) \in \rho

\therefore \rho

is symmetric Now,

\quad(1,-0 \cdot 1) \in \rho

and

(-0 \cdot 1,-9) \in \rho

but

\quad(1,-9) \notin \rho

\therefore \rho

is not transitive.

Practice Sets and Relations on Quantrex Academy →

More from Sets and Relations

Let S = \ 1, 2, 3, …, 10\ . Consider the set X = \ R : R is an equivalence relation on… JEE Advanced 2026 Consider the relation R on the set \ -2,-1,0,1,2\ defined by (a, b) ∈ R if and only if… JEE Main 2026 Let R = \ (x, y) ∈ ℕ × ℕ : _e(x + y) ≤ 2\ . Then the minimum number of elements, required… JEE Main 2026 Let A = \ 1, 4, 7\ and B = \ 2, 3, 8\ . Then the number of elements, in the relation R =… JEE Main 2026 Let A = \ 2, 3, 4, 5, 6\ . Let R be a relation on the set A × A given by (x, y) R (z, w)… JEE Main 2026 Let R be a relation defined on the set \ 1,2,3,4\ ×\ 1,2,3,4\ by R=\ ((a, b),(c, d)): 2… JEE Main 2026 Let A=\ 0,1,2, …, 9\ . Let R be a relation on A defined by (x, y) ∈ R if and only if… JEE Main 2026 Let A=\ -2,-1,0,1,2,3,4\ . Let R be a relation on A defined by x Ry if and only if 2 x+y… JEE Main 2026 The number of elements in the relation R=\ (x, y): 4 x^ 2 +y^ 2 <52, x, y ∈ Z\ is JEE Main 2026 Let S be the set of the first 11 natural numbers. Then the number of elements in A=\ B ⊆… JEE Main 2026 Let the relation R on the set M=\ 1,2,3, …, 16\ be given by R=\ (x, y): 4 y=5 x-3, x, y ∈… JEE Main 2026 Let A=\ x:|x^ 2 -10| ≤ 6\ and B=\ x:|x-2|>1\ . Then JEE Main 2026 Let A=\ 2,3,5,7,9\ . Let R be the relation on A defined by x R y if and only if 2 x ≤ 3 y… JEE Main 2026 The number of relations, defined on the set \ a, b, c, d\ , which are both reflexive and… JEE Main 2026 Let the set of all relations R on the set \ a, b, c, d, e, f\ , such that R is reflexive… JEE Advanced 2025 Let A=\ 0,1,2,3,4,5\ . Let R be a relation on A defined by (x, y) ∈ R if and only if max… JEE Main 2025 Full Sets and Relations list Sets and Relations JEE Maths PYQs JEE Mathematics PYQs All Most Important Selected Qs for JEE Advanced PYQs