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

(A) The relation is defined as $R = \{(P_{1}, P_{2}) : P_{1} \text{ and } P_{2} \text{ have the same number of sides}\}$.
$1.$ Reflexivity: For any polygon $P_{1} \in A$,$(P_{1}, P_{1}) \in R$ because $P_{1}$ has the same number of sides as itself. Thus,$R$ is reflexive.
$2.$ Symmetry: Let $(P_{1}, P_{2}) \in R$. This implies $P_{1}$ and $P_{2}$ have the same number of sides. Consequently,$P_{2}$ and $P_{1}$ have the same number of sides,so $(P_{2}, P_{1}) \in R$. Thus,$R$ is symmetric.
$3.$ Transitivity: Let $(P_{1}, P_{2}) \in R$ and $(P_{2}, P_{3}) \in R$. This means $P_{1}$ and $P_{2}$ have the same number of sides,and $P_{2}$ and $P_{3}$ have the same number of sides. Therefore,$P_{1}$ and $P_{3}$ have the same number of sides,so $(P_{1}, P_{3}) \in R$. Thus,$R$ is transitive.
Since $R$ is reflexive,symmetric,and transitive,it is an equivalence relation.
The set of all elements in $A$ related to the right-angled triangle $T$ consists of all polygons that have the same number of sides as $T$. Since $T$ has $3$ sides,the set consists of all triangles in $A$.