Show that the relation $R$ defined in the set $A$ of all polygons as $R=\left\{\left(P_{1}, P_{2}\right):\right.$ $P _{1}$ 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 $5 ?$

$R = \{ \left( {{P_1},{P_2}} \right):{P_1}$ and $ P _{2}$ have same the number of sides $\}$

$R$ is reflexive,

since $\left( P _{1}, \,P _{1}\right) \in R ,$ as the same polygon has the same number of sides with itself.

Let $\left( P _{1}, P _{2}\right) \in R$

$\Rightarrow P _{1}$ and $P _{2}$ have the same number of sides.

$\Rightarrow P _{2}$ and $P _{1}$ have the same number of sides.

$\Rightarrow\left( P _{2}, P _{1}\right) \in R$

$\therefore R$ is symmetric.

Now,

Let $\left( P _{1}, P _{2}\right),\left( P _{2}, P _{3}\right) \in R$

$\Rightarrow P _{1}$ 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.

$\Rightarrow\left( P _{1}, P _{3}\right) \in R$

$\therefore R$ is transitive.

Hence, $R$ is an equivalence relation.

The elements in $A$ related to the right-angled triangle $(T)$ 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.