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કોઈ ચોક્કસ સમયે કોઈ એક નગરમાં વસતા મનુષ્યોના ગણ $A$ પર વ્યાખ્યાયિત સંબંધ $R =\{(x, y): x$ ની ઊંચાઈ $y$ ની ઊંચાઈ કરતાં બરાબર $7$ સેમી વધારે છે. $\} $ સ્વવાચક, સંમિત અથવા પરંપરિત સંબંધ છે કે નહિ તે નક્કી કરો ?
Solution
$R =\{( x , y ): x$ 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.
