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

Question:

Show that the relation R, defined on 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:

We observe the following properties on R.

Reflexivity: Let $P_{1}$ be an arbitrary element of $A$.

Then, polygon $P_{1}$ and $P_{1}$ have the same number of sides, since they are one and the same.

$\Rightarrow\left(P_{1}, P_{1}\right) \in R$ for all $P_{1} \in A$

So, $R$ is reflexive on $A$.

Symmetry : 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.

$\Rightarrow\left(P_{2}, P_{1}\right) \in R$ for all $P_{1}, P_{2} \in A$

So, $R$ is symmetric on $A$.

Transitivity: 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 and $P_{2}$ and $P_{3}$ have the same number of sides.

$\Rightarrow P_{1}, P_{2}$ and $P_{3}$ have the same number of sides.

$\Rightarrow P_{1}$ and $P_{3}$ have the same number of sides.

$\Rightarrow\left(P_{1}, P_{3}\right) \in R$ for all $P_{1}, P_{3} A$

So, $R$ is transitive on $A$.

Hence, R is an equivalence relation on the set A.

Also, the set of all the triangles $\in A$ is related to the right angle triangle $T$ with the sides $3,4,5$.