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» (i) Reflexive R is reflexive if T₁ R T₁ Since T₁ ≅ T₁ ∴ R is reflexive. (ii) Symmetric R is symmetric if T₁ R T₂ ⇒ T₂ R T₁ Since T₁ ≅ T₂ ⇒ T₂ ≅ T₁ ∴ R is symmetric. (iii) Transitive R is transitive if T₁ R T₂ and T₂ R T₃ ⇒ T₁R T₃ Since T₁ ≅ T₂ and T₂ ≅ T₃ ⇒ T₁ ≅ T₃ ∴ R is transitive From (i), (ii) and (iii), we get 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»

(i) Reflexive

R is reflexive if T₁ R T₁

Since T₁ ≅ T₁

∴ R is reflexive.

(ii) Symmetric

R is symmetric if T₁ R T₂ ⇒ T₂ R T₁

Since T₁ ≅ T₂ ⇒ T₂ ≅ T₁

∴ R is symmetric.

(iii) Transitive

R is transitive if T₁ R T₂ and T₂ R T₃ ⇒ T₁R T₃

Since T₁ ≅ T₂ and T₂ ≅ T₃ ⇒ T₁ ≅ T₃

∴ R is transitive From (i), (ii) and (iii), we get R is an equivalence relation.