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निर्थारित कीजिए कि क्या निम्नलिखित संबंधों में से प्रत्येक स्वतुल्य, सममित तथा संक्रामक हैं :
किसी विशेष समय पर किसी नगर के निवासियों के समुच्चय में निम्नलिखित संबंध $R.$
$R =\{(x, y): x, y$ की पत्नी है$\}$
Solution
$R =\{( x , y ): x$ is the wife of $y \}$
Now,
$(x, x) \notin R$
since $x$ cannot be the wife of herself.
$\therefore R$ is not reflexive.
Now, let $(x, y) \in R$
$\Rightarrow x$ is the wife of $y$ Clearly y is not the wife of $x$. $\therefore $ $( y , x ) \notin R$
Indeed, if $x$ is the wife of $y$, then $y$ is the husband of $x$.
$\therefore $ $R$ is not transitive.
Let $( x , \,y ),\,( y , \,z ) \in R$
$\Rightarrow $ $x$ is the wife of $y$ and $y$ is the wife of $z$.
This case is not possible. Also, this does not imply that $x$ is the wife of $z$.
$\therefore $ $( x ,\, z ) \notin R$
$\therefore $ $R$ is not transitive.
Hence, $R$ is neither reflexive, nor symmetric, nor transitive.
