Question

Check whether the relation $R$ defined in the set $\{1,2,3,4,5,6\}$ as $R=\{(a, b): b=a+1\}$ is reflexive, symmetric or transitive.

Answer 1

$\begin{array}{l}\text { Here } R=\{(a, b): b=a+1\} \\=\{(a, a+1): a, a+1 \in\{1,2,3,4,5,6\}\} \\=\{(1,2)(2,3),(3,4),(4,5),(5,6)\} \\\text { (i) } \quad \mathrm{R} \text { is not reflexive as }(a, a) \notin R \forall \text { a. } \\\text { (ii) } R \text { is not symmetric as }(a, b) \in R \text { but (b, a) } \notin R \\\text { (iii) } R \text { is not transitive as }(a, b) \in R \text { and (a, c) } \in R \text { but }(a, c) \notin R \\ \therefore (1,2) \in R,(2,3) \in R \text { but }(1,3) \notin R \\\end{array}$