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
Correct Answer: A
Solution :
[a] \[\rho \] is reflexive, since \[\left| a-a \right|=0<\frac{1}{2}\] for all \[a\in R.\] \[\rho \] is symmetric, since \[\Rightarrow \left| b-a \right|<\frac{1}{2}\] \[\rho \] is not transitive. For. If we take three numbers \[\frac{3}{4},\frac{1}{3},\frac{1}{8}.\] Then, \[\left| \frac{3}{4}-\frac{1}{3} \right|=\frac{5}{12}<\frac{1}{2}\] and \[\left| \frac{1}{3}-\frac{1}{8} \right|=\frac{5}{24}<\frac{1}{2}\] But, \[\left| \frac{3}{4}-\frac{1}{8} \right|=\frac{5}{8}>\frac{1}{2}\] Thus, \[\frac{3}{4}\rho \frac{1}{3}\] and \[\frac{1}{3}\rho \frac{1}{8}\] but \[\frac{3}{4}(\tilde{\ }\rho )\frac{1}{8}\]
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