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JEE Main 202131 AugustMorning Shift
Exam
JEE Main
Year
2021
Date
31 August
Shift
Morning Shift
Subject
Mathematics
Chapter
Sets and Relations

JEE Main 2021 · 31 August · Morning Shift · Mathematics · Sets and Relations

Which of the following is not correct for relation R on the set of...

Which of the following is not correct for relation RR on the set of real numbers?

Options

  1. A(x,y)∈R⇔∣x∣−∣y∣≤1( x, y ) \in R ⇔ | x | - | y | \le 1 is reflexive but not symmetric.
  2. B(x,y)∈R⇔∣x−y∣≤1( x, y ) \in R ⇔ | x - y | \le 1 is reflexive and symmetric.
  3. C(x,y)∈R⇔0<∣x−y∣≤1( x, y ) \in R ⇔ 0 < | x - y | \le 1 is symmetric and transitive.
  4. D(x,y)∈R⇔0<∣x∣−∣y∣≤1( x, y ) \in R ⇔ 0 < | x | - | y | \le 1 is not transitive but symmetric.

Correct answer

D. (x,y)∈R⇔0<∣x∣−∣y∣≤1( x, y ) \in R ⇔ 0 < | x | - | y | \le 1 is not transitive but symmetric.

Step-by-step solution

We know that (x,y)∈R,(y,x)∈R⇒( x, y ) \in R, ( y, x ) \in R ⇒symmetric

(x,y)∈R,(y,z)∈R and  (z,x)∈R⇒( x, y ) \in R, ( y, z ) \in R \text{\text{ and } } ( z, x ) \in R ⇒transitive
\left0<∣x−y∣≤10<∣y−x∣≤1\begin{array}{ll}0 < | x - y | \le 1 \\ 0 < | y - x | \le 1\end{array}is symmetric

let

0<∣x−y∣≤10 < | x - y | \le 1 .......(1)
(1,2)∈R( 1, 2 ) \in R and (2,3)∈R( 2, 3 ) \in R satisfy the equation (1)
but (1,3)∉R( 1, 3 ) \notin R not satisfied
Hence $0 < | x - y | \le 1 is symmetric but not transitive relation

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