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 $y\}$
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.
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