Question

Let T be the set of all triangles in a plane with R as relation in T given by R = {(T₁, T₂) : T₁ ≅ T₂}. Show that R is an equivalence relation.

Answer 1

We have the relation, 

R = {(T₁, T₂) : T₁ ≅ T₂}

Reflexivity : As Each triangle is congruent to itself,

i.e., T₁ ≅ T₂ ∀ T₁ ∈ T

Thus, R is reflexive.

Symmetry : Let T₁, T₂ ∈ T, such that

(T_1, T₂₎ ∈ R

⇒ T₁ ≅ T₂

T₂ ≅ T₁ 

⇒ (T_2, T₁) ∈ R

i.e., R is symmetric.

Transitivity : Let T₁, T_2, T₃ ∈ T, such that (T₁, T₂) ∈ R and (T₂, T₃) ∈ R

⇒ T₁ ≅ T₂ and T₂ ≅ T₃

⇒ T₁ ≅ T₃ ⇒ (T₁, T₃) ∈ R

i.e., R is transitive.

Hence, R is an equivalence relation.