Show that the relation R defined in the set A of all polygons as R

Question:

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

Solution:

$\mathrm{R}=\left\{\left(P_{1}, P_{2}\right): P_{1}\right.$ and $P_{2}$ have same the number of sides $\}$

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

Let $\left(P_{1}, P_{2}\right) \in \mathrm{R}$.

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

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

$\Rightarrow\left(P_{2}, P_{1}\right) \in \mathrm{R}$

∴R is symmetric.

Now,

Let $\left(P_{1}, P_{2}\right),\left(P_{2}, P_{3}\right) \in \mathrm{R}$.

$\Rightarrow P_{1}$ 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.

$\Rightarrow\left(P_{1}, P_{3}\right) \in \mathrm{R}$

∴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.