# Determine whether Relation $R$ is reflexive, symmetric and transitive: Relation $R$ in the set $A$ of all human beings in a town at a particular time given by $R=\{(x,y):x\; \text{and y live in the same locality}\}$

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Toolbox:
• A relation R in a set A is called reflexive. if $(a,a) \in R\; \qquad a\in for\; every\;A$
• A relation R in a set A is called symmetric. if $(a_1,a_2) \in R\;=>\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called transitive. if $(a_1,a_2) \in\;and\; R(a_2,a_3)=>that \;(a_1,a_3)\in R for\; all\; a_1,a_2,a_3 \in A$
Given the set $A$ of all human beings in a town at a particular time and $R=\{(x,y):x \text{and y live in the same locality}\}$
If $x$ lives in a locality, $\rightarrow$ obviously $(x,x) \in R$. Hence $R$ is reflexive.
If $x$ and $y$ live in a locality, i.e., if $(x,y) \in R$ $\rightarrow$ it follows that $y\;$and $x$ live in the same locality, i.e, $(y,x) \in R$. Hence $R$ is symmetric.
If $x$ and $y$ live in a locality, i.e., if $(x,y) \in R$ and if $y$ and $z$ live in the same locality, i.e., if $(y,z) \in R$ $\rightarrow$ it follows that $x\;$and $z$ also live in the same locality, i.e, $(x,z) \in R$. Hence $R$ is transitive.