Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $R$ in the set $A$ of human beings in a town at a particular time given by

$R =\{(x, y): x$ is exactly $7\,cm $ taller than $y\}$

$R =\{( x , y ): x$ is exactly $7\,cm$ taller than $y\}$

Now, $(x, x) \notin R$

since human being $x$ cannot be taller than himself.

$\therefore R$ is not reflexive.

Now, let $(x, y) \in R$

$\Rightarrow x$ is exactly $7 \,cm$ taller than $y$.

Then, $y$ is not taller than $x$ . $[$ since, $y $ is $7$ $cm$ smaller than $x]$

$\therefore(y, \,x) \notin R$

Indeed if $x$ is exactly $7 \,cm$ taller than $y$, then $y$ is exactly $7\, cm$ shorter than $x$.

$\therefore \,R$ is not symmetric.

Now,

Let $( x , \,y ),\,( y ,\, z ) \in R$

$\Rightarrow \,x$ is exactly $7 \,cm$ taller than $y$ and $y$ is exactly $7\, cm$ taller than $z$.

$\Rightarrow \,x$ is exactly $14\, cm$ taller than $z$

$\therefore(x,\, z) \notin R$

$\therefore \,R$ is not transitive.

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