Mathematics · Sets and Relations

JEE Main 2024 — 4 April, Shift 2 — Question 4

Let a relation R on $\mathbb{N} \times \mathbb{N}$ be defined as : $\left(x_{1}, y_{1}\right) R\left(x_{2}, y_{2}\right)$ if and only if $x_{1} \leq x_{2}$ or $y_{1} \leq y_{2}$ Consider the two statements :

(I) R is reflexive but not symmetric.

(II) R is transitive

Then which one of the following is true?

Option A:
Only (II) is correct.
Option B:
Only (I) is correct.
Correct
Option C:
Both (I) and (II) are correct.
Option D:
Neither (I) nor (II) is correct.

Answer: B

Step-by-step solution

All $\left(\left(\mathrm{x}_{1} \mathrm{y}_{1}\right),\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\right)$ are in R where

$\mathrm{x}_{1}, \mathrm{y}_{1} \in \mathrm{~N}$ $\therefore \mathrm{R}$ is reflexive

$((1,1),(2,3)) \in R$

but $((2,3),(1,1)) \notin R$

$\therefore \mathrm{R}$ is not symmetric

$((2,4),(3,3)) \in \mathrm{R}$ and $((3,3),(1,3)) \in \mathrm{R}$

but $((2,4)$ , $(1,3)) \notin \mathrm{R}$

$\therefore \mathrm{R}$ is not transitive

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 2
Subject: Mathematics
Chapter: Sets and Relations
Topic: Types of Relations