First slide

Relations XII

Question

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

Easy

A

Reflexive and Symmetric
B

Symmetric only
C

Transitive only
D

Anti-symmetric only

Solution

$|a - a| = 0 < 1 ∴ aRa \forall a \in R$
$∴ R is reflexive .$
Again $aRb \Rightarrow |a - b| \leq 1 \Rightarrow |b - a| \leq 1 \Rightarrow bRa$
$∴$ R is symmetric, Again $1 R \frac{1}{2} and \frac{1}{2} R 1$ but $\frac{1}{2} \neq 1$
$∴$ R is not anti-symmetric.
Further, 1 R 2 and 2 R 3 but $1 ℝ 3, [∵ |1 - 3| = 2 > 1]$
$∴$ R is not transitive.

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