Question

Determine whether the following relation is reflexive, symmetric and transitive

Relative $R$ in the set $N$ of natural numbers defined as $R=\left\{\right(x,y):y=x+5andx<4\}$
enter 1-Reflexive
2-Symmetric
3-Transitive
4-Equivalence
5-None

Answer 1

$R=\left\{\right(x,y):y=x+5\text{and}x<4\}\Rightarrow R=\left\{\right(1,6),(2,7),(3,8\left)\right\}$

It is seen that $(1,1)\notin R\Rightarrow R$ is not reflexive.

Also $(1,6)\in R$.

But, $(1,6)\notin R$.

$\therefore R$ is not symmetric.

Now, since there is no pair in $R$ such that $(x,y)$ and $(y,z)\in R$, then $(x,z)$ cannot belong to $R$.

$\therefore R$ is transitive.

Hence, $R$ is neither reflexive, nor symmetric, but transitive.