A) Reflexive and symmetric
B) Symmetric only
C) Transitive only
D) Anti-symmetric only
Correct Answer: A
Solution :
\[\left| a-a \right|=0<1\] \[\therefore \,\,\,a\,\,R\,\,a\,\,\forall \,\,a\in R\] \[\therefore \] R is reflexive. Again \[a\,\,R\,\,b\Rightarrow \,\,\left| a-b \right|\le 1\Rightarrow \left| b-a \right|\le 1\Rightarrow b\,R\,a\] \[\therefore \] R is symmetric. Now, \[1R\frac{1}{2}\] and \[\frac{1}{2}Rl\]but \[\frac{1}{2}\ne 1\] \[\therefore \] R is not anti-symmetric. Further, \[1\,\,R\,\,2\] and \[2\,\,R\,\,3\] but \[1\cancel{R}3,\] \[\left[ \because \,|1-3|=2>1 \right]\] \[\therefore \] R is not transitive.
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