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

1.	Show that the relation R defined in the set A of all triangles as R = {(T1,T2): T2 is similar to T2}, is equivalence relation. Consider three right angle triangles T2 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8,10. Which triangles among T1, T2 and T3 are related?
Answer» (T₁, T₁) ∈ R ⇒ T₁ is similar to T₁which is true. Hence R is reflexive (T₁, T₂)∈ R ⇒ T₁ is similar to T₂ ⇒ T₂ is similar to T₁ ⇒ (T₂, T₁) ∈ 2 hence R is symmetric (T₁, T₂) ∈ R, (T₂, T₃) ∈ R ⇒ T is similar to T₂, T₂ is similar to T₃ ⇒ T₁ is similar to T₃ ⇒(T₁,T₃) ∈ R hence R is transitive ∴ R is an equivalence Relation. T₁ is similar to T₃ as hence T₁ and T₃ are related.

Show that the relation R defined in the set A of all triangles as R = {(T1,T2): T2 is similar to T2}, is equivalence relation. Consider three right angle triangles T2 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8,10. Which triangles among T1, T2 and T3 are related?

Answer»

(T₁, T₁) ∈ R ⇒ T₁ is similar to T₁which is true.

Hence R is reflexive

(T₁, T₂)∈ R ⇒ T₁ is similar to T₂

⇒ T₂ is similar to T₁

⇒ (T₂, T₁) ∈ 2

hence R is symmetric

(T₁, T₂) ∈ R, (T₂, T₃) ∈ R

⇒ T is similar to T₂, T₂ is similar to T₃

⇒ T₁ is similar to T₃ ⇒(T₁,T₃) ∈ R

hence R is transitive

∴ R is an equivalence Relation.

T₁ is similar to T₃ as hence T₁ and T₃ are related.