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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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.
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.
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.