Question

Whether following subsets of N are total ordered set for relation “x divides y or not” : 

(i) {2, 4, 6, 8,……} 

(ii) {0, 2, 4, 6,……..} 

(iii) (3, 9, 5, 15,…} 

(iv) (5, 15, 30} 

(v) {1, 2, 3, 4} 

(vi) {a, b, ab} ∀ a, b ∈ R

Answer 1

(i) Let subset A = {2, 4, 6, 8, …} ∈ N 

A relation given in N

From equation (i) putting the value of y in equation (ii) we have

Hence, R is transitive relation.

Hence, from (i), (a), (b) and (c) the given relation is a partial ordred relation.

Hence, the second rule is not followed then the given relation is not a total ordered relation. 

For this it is clear that to prove a relation is a total ordered relative must be reflexive, symmetric and transitive and also every a, b ∈ A, (a, b) ∈ R or (b, a) ∈ R or a = b is true.

Hence, the given relation and their results are:

Relation	Conclusion
(i) (2, 4, 6, 8,…}	No
(ii) {0, 2, 4, 6,…}	No
(iii) {3, 9, 5, 15,…}	No
(iv) {5, 15, 30}	Yes
(v) {1, 2, 3, 4}	No
(vi) {a, b, ab} ∀ a, b ∈ R	No