Let $R = \{(3, 3), (5, 5), (9, 9), (12, 12), (5, 12), (3, 9), (3, 12), (3, 5)\}$ be a relation on the set $A = \{3, 5, 9, 12\}.$ Then,$R$ is

Let $P(S)$ denote the power set of $S = \{1, 2, 3, \ldots, 10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $A R_1 B$ if $(A \cap B^c) \cup (B \cap A^c) = \varnothing$ and $A R_2 B$ if $A \cup B^c = B \cup A^c, \forall A, B \in P(S)$. Then:

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 $3, 4, \text{ and } 5$?

Among the relations $S = \{(a, b) : a, b \in R - \{0\}, 2 + \frac{a}{b} > 0\}$ and $T = \{(a, b) : a, b \in R, a^2 - b^2 \in Z\}$,which of the following is true?

The number of reflexive relations on a set $A$ of $n$ elements is equal to

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.