Let T be the set of all triangles in a plane with R as relation in T g

1.	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» 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.

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»

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.