Let A be the set of all triangles in a plane. Show that the relation R

1.	Let A be the set of all triangles in a plane. Show that the relation R = {(∆1, ∆2): ∆1 ~ ∆2} is an equivalence relation on A.
Answer» Reflectivity: Consider ∆ as an arbitrary element on A We know that ∆ – ∆ => (∆, ∆) ∈ R Ɐ ∆ ∈ R Hence, R is reflective. Symmetric: Consider ∆₁ and ∆₂ ∈ A where (∆₁, ∆₂) ∈ R We get (∆₁, ∆₂) ∈ R => ∆₁ ~ ∆₂ So ∆₁ ~ ∆₂ ∈ R => (∆₁, ∆₂) ∈ R Hence, R is symmetric. Transitivity: Consider ∆₁, ∆₂ and ∆₃ ∈ A where (∆₁, ∆₂) and (∆₂, ∆₃) ∈ R We get (∆₁, ∆₂) ∈ R => ∆₁ ~ ∆₂ (∆₂, ∆₃) ∈ R => ∆₂ ~ ∆₃ It can be written as (∆₁, ∆₃) ∈ R => ∆₁ ~ ∆₃ Hence, R is transitive.

Let A be the set of all triangles in a plane. Show that the relation R = {(∆1, ∆2): ∆1 ~ ∆2} is an equivalence relation on A.

Answer»

Reflectivity:

Consider ∆ as an arbitrary element on A

We know that

∆ – ∆ => (∆, ∆) ∈ R Ɐ ∆ ∈ R

Hence, R is reflective.

Symmetric:

Consider ∆₁ and ∆₂ ∈ A where (∆₁, ∆₂) ∈ R

We get (∆₁, ∆₂) ∈ R => ∆₁ ~ ∆₂

So ∆₁ ~ ∆₂ ∈ R => (∆₁, ∆₂) ∈ R

Hence, R is symmetric.

Transitivity:

Consider ∆₁, ∆₂ and ∆₃ ∈ A where (∆₁, ∆₂) and (∆₂, ∆₃) ∈ R

We get

(∆₁, ∆₂) ∈ R => ∆₁ ~ ∆₂

(∆₂, ∆₃) ∈ R => ∆₂ ~ ∆₃

It can be written as

(∆₁, ∆₃) ∈ R => ∆₁ ~ ∆₃

Hence, R is transitive.

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Let A be the set of all triangles in a plane. Show that the relation R = {(∆1, ∆2): ∆1 ~ ∆2} is an equivalence relation on A.

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