Self-Doubt

Question

Is the Poset (Q,Less than or equal to) a well ordered set?

Where Q denotes set of all rational numbers and relation R is less than or equal to.

Answer 1

For a toset to be a woset : 

a) Each element is discrete in nature.

b) There must be some lowermost element.If we talk in Hasse diagram analogy , the corresponding Hasse diagram must have some finite starting point to begin with.

First of all the given poset : { set of all rational numbers under <= operation } is a toset because any two rational numbers can be compared using <= operation and hence related to each other which is needed for being a toset..

But as this is a set of rational numbers , it will go till -infinity which violates point b).Moreover the elements are not of the discrete nature as between any 2 rational numbers , we have countless rational numbers ..We can find by dividing the sum of 2 elements by 2 which will be a rational number as well..

Instead , if the set were set of positive integers Z⁺ and using the same <= operation , then the corresponding toset would have been woset..In this case it wont be a woset and merely a toset..