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show that the relation r defined in the set a of a...

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:

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.

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