Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time is given by:
$R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$
Relation $R$ in the set $A$ of human beings in a town at a particular time is given by:
$R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$
(A) $R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$
$1.$ Reflexive: For any human being $x \in A$,$x$ lives in the same locality as $x$. Thus,$(x, x) \in R$. Therefore,$R$ is reflexive.
$2.$ Symmetric: Let $(x, y) \in R$. This means $x$ and $y$ live in the same locality. It follows that $y$ and $x$ also live in the same locality. Thus,$(y, x) \in R$. Therefore,$R$ is symmetric.
$3.$ Transitive: Let $(x, y) \in R$ and $(y, z) \in R$. This means $x$ and $y$ live in the same locality,and $y$ and $z$ live in the same locality. Consequently,$x$ and $z$ must live in the same locality. Thus,$(x, z) \in R$. Therefore,$R$ is transitive.
Conclusion: The relation $R$ is reflexive,symmetric,and transitive.
$1.$ Reflexive: For any human being $x \in A$,$x$ lives in the same locality as $x$. Thus,$(x, x) \in R$. Therefore,$R$ is reflexive.
$2.$ Symmetric: Let $(x, y) \in R$. This means $x$ and $y$ live in the same locality. It follows that $y$ and $x$ also live in the same locality. Thus,$(y, x) \in R$. Therefore,$R$ is symmetric.
$3.$ Transitive: Let $(x, y) \in R$ and $(y, z) \in R$. This means $x$ and $y$ live in the same locality,and $y$ and $z$ live in the same locality. Consequently,$x$ and $z$ must live in the same locality. Thus,$(x, z) \in R$. Therefore,$R$ is transitive.
Conclusion: The relation $R$ is reflexive,symmetric,and transitive.
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