Question

Let R be the relation on the set R of all real numbers defined by a R biff$| a - b | \leq 1$. Then R is

• A. Reflexive and Symmetric
• B. Symmetric only
• C. Transitive only
• D. Anti-symmetric only

Answer 1

Answer: (A)

Reflexive and Symmetric

Answer 2

$| a - a | = 0 < 1$ $\therefore a R a \forall a \in R$

$\therefore$ R is reflexive, Again a R b

⇒ $| a - b | \leq 1 \Rightarrow b - a \mid \leq 1 \Rightarrow b R a$ $\therefore$ R is symmetric,

Again $\frac { 1 } { 2 } R 1$ but

$\therefore$ R is not anti-symmetric

Further, 1 R 2 and 2 R 3 but 1 R 3 [$\because | 1 - 3 | = 2 > 1$]

$\therefore$ R is not transitive.