# Q.1 Determine whether each of the following relations are reflexive, symmetric and transitive:(v) Relation R in the set A of human beings in a town at a particular time given by(e) $R = \{(x, y) : x \;is \;father \;of \;y \}$

$R = \{(x, y) : x \;is \;father \;of \;y \}$

$(x, y) \in R$ means $x \;is \;father \;of \;y$ than $x \;cannot \, be \;father \;of \;x$  i.e. $(x, x) \notin R$.So, it is not reflexive..

$(x, y) \in R$ means $x \;is \;father \;of \;y$ than  $y \;cannot \, be \;father \;of \;x$ i.e. $(y, x) \notin R$.So, it is not symmetric.

Let, $(x, y),\left ( y,z \right )\in R$ means $x \;is \;father \;of \;y$ and $y \;is \;father \;of \;z$ than $x \;cannot \, be \;father \;of \;z$ i.e. $(x, z) \notin R$.

So, it is not transitive.

Hence, it is neither reflexive, nor symmetric and
nor transitive.

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