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

(d) R = \{(x, y) : x\;is\;wife\;of\;y\}

Answers (1)

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

\left ( x,y \right ) \in R means x\;is\;wife\;of\;y but  x\;is\;not\, wife\;of\;x i.e.\left ( x,x \right ) \notin R.

So, it is not reflexive.

\left ( x,y \right ) \in R means x\;is\;wife\;of\;y but  y\;is\;not\, wife\;of\;x i.e.\left ( y,x \right ) \notin R.

So, it is not symmetric.

Let, \left ( x,y \right ),\left ( y,z \right ) \in R means x\;is\;wife\;of\;y and y\;is\;wife\;of\;z.

This case is not possible so it is not transitive.

Hence, it is not  reflexive, symmetric and
transitive.

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