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Question
Let P be the set of all triangles in a plane and R be the relation defined on P as aRb if a is similar to b. Prove that R is an equivalence relation
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Solution
Given P = the set of all triangles in a plane.
R is the relation defined by aRb if a is similar to b.
R = {(a, b) : a is similar to b for a, b ∈ p}
(a) Reflexive:
(a, a) ⇒ a is similar to a for all a ∈ P
∴ R is reflexive.
(b) Symmetric:
Let (a, b) ∈ R ⇒ a is similar to b
⇒ b is similar to a
∴ (b, a) ∈ R
Hence R is symmetric.
c) Transitive:
Let (a, b) and (b, c) ∈ R
(a, b) ∈ R ⇒ a is similar to b
(b, c) ∈ R ⇒ b is similar to c
∴ a is similar to c.
Hence R is transitive.
∴ R is an equivalence relation on P.
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