Let T be the set of all triangles in a plane and R is a relations on T

1.	Let T be the set of all triangles in a plane and R is a relations on T defined as R = {(T1,T2): T1 is similar to T2 where T1, T2 ∈ T} then relation R is a/an 1. Only reflexive2. Only symmetric3. Only transitive4. Equivalence relation
Answer» Correct Answer - Option 4 : Equivalence relation Concept: A relation R in a set A is called Reflexive, if (a, a) ∈ R, for every a ∈ A. Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for all a, b ∈ A. Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈ A. A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive. Calculation: Given: R = {(T1,T2): T1 is similar to T₂ where T₁, T₂ ∈ T} and T is the set of all triangles in a plane Reflexive: As we know that, every triangle is similar to itself, so (T₁, T₁) ∈ R ∀ T₁ ∈ T Hence, relation R is reflexive. Symmetric: Suppose if (T₁, T₂) ∈ R ⇒T₁ is similar to T₂ ⇒T₂ is also similar to T₁ ⇒ (T₂, T₁) ∈ R. Hence, relation R is symmetric. Transitive: Now suppose, (T₁, T₂), (T₂, T₃) ∈ R ⇒T₁ is similar to T₂ and T₂ is similar to T₃ ⇒T₁ is similar to T₃ . So (T₁, T₃) ∈ R. Hence, relation R is transitive. Hence, relation R is an equivalence relation.

Let T be the set of all triangles in a plane and R is a relations on T defined as R = {(T1,T2): T1 is similar to T2 where T1, T2 ∈ T} then relation R is a/an 1. Only reflexive2. Only symmetric3. Only transitive4. Equivalence relation

Answer» Correct Answer - Option 4 : Equivalence relation

Concept:

A relation R in a set A is called

Reflexive, if (a, a) ∈ R, for every a ∈ A.
Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for all a, b ∈ A.
Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b, c ∈ A.

A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Calculation:

Given: R = {(T1,T2): T1 is similar to T₂ where T₁, T₂ ∈ T} and T is the set of all triangles in a plane

Reflexive:

As we know that, every triangle is similar to itself, so (T₁, T₁) ∈ R ∀ T₁ ∈ T

Hence, relation R is reflexive.

Symmetric:

Suppose if (T₁, T₂) ∈ R ⇒T₁ is similar to T₂ ⇒T₂ is also similar to T₁ ⇒ (T₂, T₁) ∈ R.

Hence, relation R is symmetric.

Transitive:

Now suppose, (T₁, T₂), (T₂, T₃) ∈ R ⇒T₁ is similar to T₂ and T₂ is similar to T₃ ⇒T₁ is similar to T₃ . So (T₁, T₃) ∈ R.