Show that the relation R defined in the set *A* of all polygons as R = {(*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?

#### Solution

R = {(*P*_{1}, *P*_{2}): *P*_{1} and *P*_{2} have same the number of sides}

R is reflexive since (*P*_{1}, *P*_{1}) ∈ R as the same polygon has the same number of sides with itself.

Let (*P*_{1}, *P*_{2}) ∈ R.

⇒ *P*_{1} and *P*_{2} have the same number of sides.

⇒ *P*_{2} and *P*_{1} have the same number of sides.

⇒ (*P*_{2}, *P*_{1}) ∈ R

∴R is symmetric.

Now,

Let (*P*_{1}, *P*_{2}), (*P*_{2}, *P*_{3}) ∈ R.

⇒ *P*_{1} and *P*_{2} have the same number of sides. Also, *P*_{2} and *P*_{3} have the same number of sides.

⇒ *P*_{1} and *P*_{3} have the same number of sides.

⇒ (*P*_{1}, *P*_{3}) ∈ 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.