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
$R =\{(x, y): x$ and $y$ live in the same locality $\}$
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
$R =\{(x, y): x$ and $y$ live in the same locality $\}$
$R =\{( x , y ): x$ and $y $ live in the same locality $\}$
Clearly, $( x , x ) \in R$ as $x$ and $x$ is the same human being.
$\therefore R$ is reflexive.
If $(x, y) \in R,$ then $x$ and $y$ live in the same locality.
$\Rightarrow y$ and $x$ live in the same locality.
$\Rightarrow(y, x) \in R$
$\therefore R$ is symmetric.
Now, let $(x, y) \in R$ and $(y, z) \in R$
$\Rightarrow x$ and $y$ live in the same locality and $y$ and $z$ live in the same locality.
$\Rightarrow x$ and $z$ live in the same locality.
$\Rightarrow(x, z) \in R$
$\therefore R$ is transitive.
Hence, $R$ is reflexive, symmetric and transitive.
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