Let f ∶ x → y be a function. Define a relation R in x given

1.	Let f ∶ x → y be a function. Define a relation R in x given by R = {(a, b): (a) = (b)}. Examine whether R is an equivalence relation or not.
Answer» We have relation on defined by R = {(a, b): f(a) = f(b)}, where f ∶ X → Y be a function. Reflexivity : Let a ∈ X. Since, f(a) = f(a) ⇒ (a, a) ∈ R ∀ a ∈ X. Hence, relation R is a reflexive relation on set . Symmetricity : Let a, b ∈ X such that (a, b) ∈ R. ⇒ f(a) = f(b)⇒ f(b) = f(a) ∀ a, b ∈ X. Hence, relation R is a symmetric relation on set X. Transitivity : Let a, b, c ∈ X such that (a,b) ∈ R and (b,c) ∈ R. ⇒ f(a) = f(b) and f(b) = f(c) ⇒ f(a) = f(c) ⇒ (a, c) ∈ R. Hence, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R∀ a, b, c ∈ X Hence, relation R is a transitive relation on set X. Since, relation R is reflexive, symmetric and transitive relation on set X. Therefore, relation R is an equivalence relation.

Let f ∶ x → y be a function. Define a relation R in x given by R = {(a, b): (a) = (b)}. Examine whether R is an equivalence relation or not.

Answer»

We have relation on defined by R = {(a, b): f(a) = f(b)}, where f ∶ X → Y be a function.

Reflexivity : Let a ∈ X. Since, f(a) = f(a) ⇒ (a, a) ∈ R ∀ a ∈ X.

Hence, relation R is a reflexive relation on set .

Symmetricity : Let a, b ∈ X such that (a, b) ∈ R.

⇒ f(a) = f(b)⇒ f(b) = f(a) ∀ a, b ∈ X.

Hence, relation R is a symmetric relation on set X.

Transitivity : Let a, b, c ∈ X such that (a,b) ∈ R and (b,c) ∈ R.

⇒ f(a) = f(b) and f(b) = f(c)

⇒ f(a) = f(c)

⇒ (a, c) ∈ R.

Hence, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R∀ a, b, c ∈ X

Hence, relation R is a transitive relation on set X.

Since, relation R is reflexive, symmetric and transitive relation on set X.

Therefore, relation R is an equivalence relation.

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