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.
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.
Let $A =\{1,2,3,4,5,6\}$.
A relation $R$ is defined on set $A$ as: $R=\{(a, b): b=a+1\}$
$\therefore R =\{(1,2),(2,3),(3,4),(4,5),(5,6)\}$
we can find $(a, a) \notin R,$ where $a \in A$
For instance,
$(1,1),\,(2,2),\,(3,3),\,(4,4),\,(0,5),\,(0,6) \notin R$
$\therefore R$ is not reflexive.
It can be observed that $(1,2) \in R ,$ but $(2,1)\notin R$
$\therefore R$ is not symmetric.
Now, $(1,2),\,(2,3) \in R$
But, $(1,3)\notin R$
$\therefore R$ is not transitive
Hence, $R$ is neither reflexive, nor symmetric, nor transitive.
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