Question

A relation R is defined in a set N of natural numbers such that aRb if a is a divisor of b, Prove that R is a partially ordered relation but not a total ordered relation.

Answer 1

Given set N = {1, 2, 3, 4,…} = Set of natural numbers 

Where a relation is define as 

aRb ⇔ a is divisor of b ∀ a, b ∈ N 

There R is a partial ordered relation 

if it is reflexive, anti symmetric and transitive. 

(i) Reflexivity: Let a ∈ N 

a ∈ N ⇒ a is divisor of a ⇒ (a, a) ∈ R ∀ a ∈ N R is reflexive relation. 

(ii) Anti Symmetricity: 

Let a, b ∈ N is in this way (a, b) ∈ R 

(a, b) ∈ R ⇔ a is divisor of b. 

⇔ b is disivor of a. 

if a = b ∀ a, b ∈ N 

⇔ (b, a) ∈ R; a = b, ∀ a, b ∈ N

(a, b) ∈ R ⇔ (b, a) ∈ R 

⇒ a = b ∀ a, b ∈ N 

So, R is anti-symmetric relation. 

(iii) Transitivity: 

Let a, b, c ∈ N is in this way 

(a, b) ∈ R and (b, c) ∈ R

So, (a, b) ∈ R, (b, c) ∈ R 

⇒ (a, c) ∈ R ∀ a, b, c ∈ N 

So, R is a transitive relation. 

Hence, from (i), (ii) and (iii) the given relation is a partially ordered relation. 

Hence Proved.