Question

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

Answer 1

Let $A=\{1,2,3,4,5,6\}$.
A relative $R$ is defined on set $A$ as:
$R=\left\{\right(a,b):b=a+1\}$
$\therefore R=\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$
We can find $(a,a)\notin R$, where $a\in A$.
For instance, $(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\notin R$
$\therefore R$ 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.