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 \}

Answers (1)

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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