Answer to Question #238817 in Discrete Mathematics for Sudhansu

Question

For divisibility relation on the set {1,2,3,6,8,12,24,36}, draw Hasse diagram. Then find minimal, maximal, greatest and least elements. Then give the topological sort using the divisibility relation

Answer 1

Solution:

Given: A = {1,2,3,6,8,12,24,36, /}

Hasse diagram:

Maximal elements are 36 and 24 since they are succeeding all the elements. 

Minimal element is 1 since it is preceding all the elements. 

Greatest element does not exist since there is no any one element that succeeds all the elements. 

Least element is 1 since there is no any one element that precedes 1. 

If A is a poset with partial order ≤, we sometimes need to find a linear order < for the set A that will merely be an extension of the given partial order in the sense that if a ≤ b, then a < b. The process of constructing a linear order such as is called Toplogical Sorting .

Now, topological sorting:

Type 1 = {36,24,8,12,6,3,2,1}

Type 2 = {36,24,8,12,6,2,3,1}