Let Z be the set of all integers and R be relation on Z defined as R =

1.	Let Z be the set of all integers and R be relation on Z defined as R = {(a, b) : a, b ∈ Z and is divisible by 5}. Prove that R is an equivalence relation.
Answer» Given, R = {(a, b) : a, b ∈ Z and (a – b) is divisible by 5} Reflexivity : ∀ a ∈ Z a – a = 0 is divisible by 5 ⇒ (a, a) ∈ R ∀ a ∈ Z Hence, R is reflexive. Symmetry : Let (a, b) ∈ R ⇒ a – b is divisible by 5 ⇒ – (b – a) is divisible by 5 ⇒ (b – a) is divisible by 5 ⇒ (b, a) ∈ R Hence, R is symmetric. Transitivity : Let (a, b), (b, c) ∈ R ⇒ (a – b) and (b – c) are divisible by 5 ⇒ (a – b + b – c) is divisible by 5 ⇒ a – c is divisible by 5 ⇒ (a, c) ∈ R Thus, R is an equivalence relation.

Let Z be the set of all integers and R be relation on Z defined as R = {(a, b) : a, b ∈ Z and is divisible by 5}. Prove that R is an equivalence relation.

Answer»

Given,

R = {(a, b) : a, b ∈ Z and (a – b) is divisible by 5}

Reflexivity : ∀ a ∈ Z

a – a = 0 is divisible by 5

⇒ (a, a) ∈ R ∀ a ∈ Z

Hence, R is reflexive.

Symmetry : Let (a, b) ∈ R

⇒ a – b is divisible by 5

⇒ – (b – a) is divisible by 5

⇒ (b – a) is divisible by 5

⇒ (b, a) ∈ R

Hence, R is symmetric.

Transitivity : Let (a, b), (b, c) ∈ R

⇒ (a – b) and (b – c) are divisible by 5

⇒ (a – b + b – c) is divisible by 5

⇒ a – c is divisible by 5

⇒ (a, c) ∈ R

Thus, R is an equivalence relation.