• # question_answer 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] A) Reflexive and Symmetric B) Symmetric only C) Transitive only D) Anti-symmetric only

$|a-a|=0<1$ $\therefore \,a\,R\,a\,\forall \,a\in R$ $\therefore$ 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.