Question

67. Let \( R \) be the relation on the set \( R \) of all real numbers defined by \( a R b \) iff \( |a-b| \leq 1 \). The \( R \) is (a) reflexive and symmetric (b) symmetric only (c) transitive only (d) anti-symmetric only.

Answer 1

Answer: Reflexive and symmetric
Reflexive

aRa ⇒ |a – a| = 0 < 1

R is reflexive.

Symmetric

aRb ⇒ |a – b| ≤ 1 ⇒|b – a| ≤ 1 ⇒ bRa

R is symmetric.

R is not antisymmetric.

1R2, 2R3 but 1R3

Since 2 > 1,

R is not transitive.