Let n be a fixed positive integer. Define a relation R on Z as f

1.	Let n be a fixed positive integer. Define a relation R on Z as follows:(a, b) ∈ R ⇔ a − b is divisible by n. Show that R is an equivalence relation on Z.
Answer» Given as (a, b) ∈ R ⇔ a − b is divisible by n is a relation R defined on Z. To prove equivalence relation, the given relation should be reflexive, symmetric and transitive. We have to check these properties on R. Reflexivity: Let a ∈ N Here, a − a = 0 = 0 × n ⇒ a − a is divisible by n ⇒ (a, a) ∈ R ⇒ (a, a) ∈ R for all a ∈ Z Therefore, R is reflexive on Z. Symmetry: Let (a, b) ∈ R Here, a − b is divisible by n ⇒ a − b = np for some p ∈ Z ⇒ b − a = n(−p) ⇒ b − a is divisible by n [ p ∈ Z⇒ − p ∈ Z] ⇒ (b, a) ∈ R So, R is symmetric on Z. Transitivity: Let (a, b) and (b, c) ∈ R Here, a − b is divisible by n and b − c is divisible by n. ⇒ a − b = np for some p ∈ Z And b − c = nq for some q ∈ Z a – b + b − c = np + nq ⇒ a − c = n(p + q) ⇒ (a, c) ∈ R for all a, c ∈ Z So, R is transitive on Z. ∴ R is reflexive, symmetric and transitive. Hence, R is an equivalence relation on Z.

Let n be a fixed positive integer. Define a relation R on Z as follows:(a, b) ∈ R ⇔ a − b is divisible by n. Show that R is an equivalence relation on Z.

Answer»

Given as (a, b) ∈ R ⇔ a − b is divisible by n is a relation R defined on Z.

To prove equivalence relation, the given relation should be reflexive, symmetric and transitive.

We have to check these properties on R.

Reflexivity:

Let a ∈ N

Here, a − a = 0 = 0 × n

⇒ a − a is divisible by n

⇒ (a, a) ∈ R

⇒ (a, a) ∈ R for all a ∈ Z

Therefore, R is reflexive on Z.

Symmetry:

Let (a, b) ∈ R

Here, a − b is divisible by n

⇒ a − b = np for some p ∈ Z

⇒ b − a = n(−p)

⇒ b − a is divisible by n [ p ∈ Z⇒ − p ∈ Z]

⇒ (b, a) ∈ R

So, R is symmetric on Z.

Transitivity:

Let (a, b) and (b, c) ∈ R

Here, a − b is divisible by n and b − c is divisible by n.

⇒ a − b = np for some p ∈ Z

And b − c = nq for some q ∈ Z

a – b + b − c = np + nq

⇒ a − c = n(p + q)

⇒ (a, c) ∈ R for all a, c ∈ Z

So, R is transitive on Z.

∴ R is reflexive, symmetric and transitive.

Hence, R is an equivalence relation on Z.