Functions Question 456
Question: Let $ \rho $ be the relation on the set R of all real numbers defined by setting $ a\rho b $ if f $ | a-b |\le \frac{1}{2}. $ then, $ \rho $ is
Options:
A) Reflexive and symmetric but not transitive
B) Symmetric and transitive but not reflexive
C) Transitive but neither reflexive nor symmetric
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ \rho $ is reflexive, since $ | a-a |=0<\frac{1}{2} $ for all $ a\in R. $ $ \rho $ is symmetric, since
$ \Rightarrow | b-a |<\frac{1}{2} $ $ \rho $ is not transitive. For. If we take three numbers $ \frac{3}{4},\frac{1}{3},\frac{1}{8}. $ Then, $ | \frac{3}{4}-\frac{1}{3} |=\frac{5}{12}<\frac{1}{2} $ and $ | \frac{1}{3}-\frac{1}{8} |=\frac{5}{24}<\frac{1}{2} $ But, $ | \frac{3}{4}-\frac{1}{8} |=\frac{5}{8}>\frac{1}{2} $ Thus, $ \frac{3}{4}\rho \frac{1}{3} $ and $ \frac{1}{3}\rho \frac{1}{8} $ but $ \frac{3}{4}(\tilde{\ }\rho )\frac{1}{8} $
BETA
