JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Sets Relations and Functions

Which of the following is not correct for relation

R

on the set of real numbers?

(A)
$(x, y) \in R \Leftrightarrow 0 < ∣ x ∣ - ∣ y ∣ \leq 1$
is neither transitive nor symmetric.
(B)
$(x, y) \in R \Leftrightarrow ∣ x - y ∣ \leq 1$
is reflexive and symmetric.
(C)
$(x, y) \in R \Leftrightarrow ∣ x ∣ - ∣ y ∣ \leq 1$
is reflexive but not symmetric.
(D)
$(x, y) \in R \Leftrightarrow 0 < ∣ x - y ∣ \leq 1$
is symmetric and transitive.✓ Correct

Answer: (D)

Solution

The relation

0 < ∣ x - y ∣ \leq 1

is symmetric, since

∣ x - y ∣ = ∣ y - x ∣

, but it is not transitive: taking

x = 0, y = 0.6, z = 1.2

, both

∣ x - y ∣ = 0.6

and

∣ y - z ∣ = 0.6

lie in

(0, 1]

, yet

∣ x - z ∣ = 1.2 > 1

. So the statement that

0 < ∣ x - y ∣ \leq 1

is symmetric and transitive is the one that is not correct.

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option