Question

Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\right(P_1,P_2):P_1$ and $P_2$ have same number of sides$\}$, is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3,4$ and $5$?

Answer 1

$R=\left\{\right(P_1,P_2):P_1\text{and}{P}_2\text{have same the number of sides}\}$
$R$ is reflexive since $(P_1,P_1)\in R$ as the same polygon has the same number of sides with itself.
Let $(P_1,P_2)\in R$.
$\Rightarrow P_1$ and $P_2$ have the same number of sides.
$\Rightarrow P_2$ and $P_1$ have the same number of sides.
$\Rightarrow (P_2,P_1)\in R$
$\therefore R$ is symmetric.
Now,
Let $(P_1,P_2),(P_2,P_3)\in R$.
$\Rightarrow P_1$ and $P_2$ have the same number of sides. Also, $P_2$ and $P_3$ have the same number of sides.
$\Rightarrow P_1$ and $P_3$ have the same number of sides.
$\Rightarrow (P_1,P_3)\in R$
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.
The elements in $A$ related to the right-angled triangle $\left(T\right)$ with sides $3,4,$ and $5$ are those polygons which have $3$ sides (since $T$ is a polygon with $3$ sides).
Hence, the set of all elements in $A$ related to triangle $T$ is the set of all triangles.