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

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Toolbox:
• A relation R in a set A is called reflexive. if $(a,a) \in R\;$ for every$\;a \in A$
• 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 $(a_1,a_2) \in\; R$ and $(a_2,a_3)\in R=>\;(a_1,a_3)\in R for\; all\; a_1,a_2,a_3 \in A$
Given the set $A$ of all human beings in a town at a particular time and $R=\{(x,y):\; \text{x is the father of y}\}$ :
Since $x$ cannot be his own father $\; (x,x) \not \in R.$ Hence $R$ is not reflexive.
If $x$ is the father of $y$ and $y$ cannot be the father of $x$, but his son i.e, $(x,y) \in R$ but $(y,x) \not \in R$. Hence $R$ is not symmetric.
If $x$ is the father of $y$ and $y$ is the father of $z$ then $x$ cannot be the father of $z$ but his grandfather instead.Hence $R$ is not transitive.