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 $5 ?$
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.
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