Let $R$ be a transitive relation on a set $A$ and $I$ be the identity relation on $A$. Then:

The number of relations on the set $A = \{1, 2, 3\}$ containing at most $6$ elements including $(1, 2)$,which are reflexive and transitive but not symmetric,is . . . . . . .

In a plane,two points $P$ and $Q$ are related if $OP = OQ$,where $O$ is a fixed point. The relation is:

Let $A$ be the set of all students of a boys' school. Show that the relation $R$ in $A$ given by $R = \{(a, b) : a \text{ is sister of } b\}$ is the empty relation and $R^{\prime} = \{(a, b) : \text{the difference between heights of } a \text{ and } b \text{ is less than } 3 \text{ meters}\}$ is the universal relation.

Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.

Let $Z$ be the set of all integers,$A = \{(x, y) \in Z \times Z : (x-2)^{2} + y^{2} \leq 4\}$,$B = \{(x, y) \in Z \times Z : x^{2} + y^{2} \leq 4\}$,and $C = \{(x, y) \in Z \times Z : (x-2)^{2} + (y-2)^{2} \leq 4\}$. If the total number of relations from $A \cap B$ to $A \cap C$ is $2^{p}$,then the value of $p$ is: