First slide

Relations XII

Question

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

Moderate

A

Reflexive and symmetric
B

Symmetric only
C

Transitive only
D

Anti-symmetric only

Solution

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

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