Sets Relations And Functions Question 53
Question: Let R be the relation on the set R of all real numbers defined by a R b iff $ |a-b|\le 1 $ . Then R is
[Roorkee 1998]
Options:
A) Reflexive and Symmetric
B) Symmetric only
C) Transitive only
D) Anti-symmetric only
Show Answer
Answer:
Correct Answer: A
Solution:
-
$ |a-a|=0<1 $
$ \therefore ,a,R,a,\forall ,a\in R $
$ \therefore $ R is reflexive. Again a R b
Þ $ |a-b|\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa $
$ \therefore $ R is symmetric, Again $ 1R\frac{1}{2} $ and $ \frac{1}{2}R1 $ but $ \frac{1}{2}\ne 1 $
$ \therefore $ R is not anti-symmetric. Further, 1 R 2 and 2 R 3 but $ 1,\not{R},3 $ , [ $ \because ,|1-3|=2>1 $ ]
$ \therefore $ R is not transitive.
BETA
