Question

Show that the relation R defined in the set A of all polygons as R = {(P₁, P₂): P₁ and P₂ 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?

Answer 1

R = {(P₁, P₂): P₁ and P₂ have same the number of sides}
R is reflexive since (P₁, P₁) ∈ R as the same polygon has the same number of sides with
itself.

Let (P₁, P₂) ∈ R.
⇒ P₁ and P₂ have the same number of sides.
⇒ P₂ and P₁ have the same number of sides.
⇒ (P₂, P₁) ∈ R
∴R is symmetric.

Now,
Let (P₁, P₂), (P₂, P₃) ∈ R.
⇒ P₁ and P₂ have the same number of sides. Also, P₂ and P₃ have the same number of
sides.
⇒ P₁ and P₃ have the same number of sides.
⇒ (P₁, P₃) ∈ 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.