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

Tiansitive only
D

Anti-symmetric only

Solution

$F o r a n y r e a l n u m b e r a, |a - a| \leq 1, s o t h a t (a, a) i s a n e l e m e n t o f r e l a t i o n, f o r a n y r e a l n u m b e r a$
$H e n c e, R i s r e f l e x i v e$
$S u p p o s e t h a t (a, b) \in R i t i m p l i e s t h a t |a - b| \leq 1 h e n c e, |b - a| \leq 1 T h e r e f o r e, (b, a) a l s o a n e l o e m e n t i n t h e r e l a t i o n$
$S o R i s s y m m e t r i c b u t R i s n o t t r a n s i t i v e b e c a s u e (- \frac{1}{2}, 0) \in R, (0, 1) \in R d o e s n o t i m p l i e s t h a t (- \frac{1}{2}, 1) \in R$

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