Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{N}$ of natural numbers defined as
$\mathrm{R}=\{(x, y): y=x+5 $ and $ x<4\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{N}$ of natural numbers defined as
$\mathrm{R}=\{(x, y): y=x+5 $ and $ x<4\}$
$\mathrm{R} =\{( x , y ): y = x +5$ and $ x <4\}=\{(1,6),(2,7),(3,8)\}$
It is clear that $(1,1)\notin \mathrm{R}$
$\therefore $ $\mathrm{R}$ is not reflexive.
$(1,6) \in \mathrm{R}$ But, $(1,6)\notin \mathrm{R}$
$\therefore $ $\mathrm{R}$ is not symmetric.
Now, since there is no pair in $\mathrm{R}$ such that $( \mathrm{x} , \,\mathrm{y} )$ and $( \mathrm{y} ,\, \mathrm{z} ) \in \mathrm{R} ,$ then $( \mathrm{x} ,\, \mathrm{z} )$ cannot belong to $\mathrm{R}$.
$\therefore \mathrm{R}$ is not transitive.
Hence, $\mathrm{R}$ is neither reflexive, nor symmetric, nor transitive.
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