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

Let $R = \{(1,2), (2,3), (3,3)\}$ be a relation defined on the set $A = \{1, 2, 3, 4\}$. Then the minimum number of elements needed to be added to $R$ so that $R$ becomes an equivalence relation is:

Show that the relation $R$ defined in the set $A$ of all polygons as $R = \{(P_{1}, P_{2}) : P_{1} \text{ and } P_{2} \text{ have the same number of sides}\}$,is an equivalence relation. What is the set of all elements in $A$ related to the right-angled triangle $T$ with sides $3, 4, \text{ and } 5$?

Let $X = R \times R$. Define a relation $R$ on $X$ as: $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$. Statement-$I$: $R$ is an equivalence relation. Statement-$II$: For some $(a, b) \in X$,the set $S = \{(x, y) \in X : (x, y) R (a, b)\}$ represents a line parallel to $y = x$. In the light of the above statements,choose the correct answer from the options given below:

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.

How many reflexive relations are there on a set with $3$ elements?