Well ordered Set

Question

I understood the Concept of well ordered Set.But i am little bit confused in finding the least element.Can please anybody clarify the concept of finding the least element in a Poset with an example.

Comments

Natural Numbers are well ordered. Read this.

http://mathonline.wikidot.com/the-well-ordering-principle-of-the-natural-numbers

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http://ndp.jct.ac.il/tutorials/discrete/node34.html

Go to the above link if you still dont understand ask me.

Answer 1

A well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a well-ordered set.

For example, We know that set $S = [0,1]$ is a Total-ordered set ( Assuming Relation defined on S is $ \leq $) as it is a Poset and every pair of elements are related(comparable) to each other.  Now Since it is a Total-ordered set, All  that we need to find whether it is a Well-ordered set or not, is that every non-empty subset of S must have a least element in this ordering.

But it is Not a Well-Ordered set as We can have at least one Subset of this Set which does not not any Least element.  For example,  What is the Least Element of $(0,1)$ ? There is no least element as whatever element you will say is least element, I can give you a number less than that in this set $(0,1)$. We can have more such subsets (like $(0.3,1)$) but having at least such subset of $S$ which doesn't have any least element is sufficient for it Not being Well-ordered set.

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Taking next example, Let's see whether  $(I,<=)$ where I is set of integers is it a well order set or not  and why?

It's Not a Well-ordered set. Because Integers itself doesn't have any least element. What is the least element of Integers?  or What is the least element of $(-∞, 0)$?  ..Hence, $(I, \leq)$ is Not a Well-ordered set. 

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Note that Set of positive integers Or Set of Non-negative integers Or any set of integers $[n,∞)$ where $n$ is a specific(constant/fixed) integer value... All are Well-Ordered sets as every non-empty subset of these Sets has a least element.

Comments

@Deepakk Poonia (Dee) sir

@Arjun sir

here least element means "The element to which no element is below it in Hasse diagram" OR "The one with least value in that subset" ?

eg  $\left [ Z^{-},\geq \right ]$ Hasse dia will be like -1 R - 2 as -1 > -2

.-inf

.

.

-4

-3

-2

-1

sir here if we consider subset {-2,-3,-4} least element will be -2 not -4 right ..i think it will be Well ordered also

correct me if wrong