The probability that a relation $R$ from $\{ x , y \}$ to $\{ x , y \}$ is both symmetric and transitive, is equal to

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation $R$ in the set $A$ of human beings in a town at a particular time given by

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

Let $A=\{1,2,3\} .$ Then show that the number of relations containing $(1,2) $ and $(2,3)$ which are reflexive and transitive but not symmetric is four.

In order that a relation $R$ defined on a non-empty set $A$ is an equivalence relation, it is sufficient, if $R$

Show that the number of equivalence relation in the set $\{1,2,3\} $ containing $(1,2)$ and $(2,1)$ is two.

Let $H$ be the set of all houses in a village where each house is faced in one of the directions, East, West, North, South. Let $R = \{ (x,y)|(x,y) \in H \times H$ and $x, y$ are faced in same direction $\}$ . Then the relation $' R '$ is