well ordered set

Question

The standard ordering ≤ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element..plz explain the abovev example...

Answer 1

For my answer , i am assuming you know what a well ordered set is ,

A nonempty set S ⊆ R is said to be well-ordered if every nonempty subset S has a least element.

For example ---> take a set of non negative integers {0,1,2,3,4,5.....} is a well ordered set and 0 is the least element (this must be unique)

and according to your question, we see

The nonempty subset (0, 1) of S has no least element because as you take 0.001, then i will say 0.000001 is the least, then you will argue, why not 0.00000000001 is the least 

We can say that the set of real numbers  [0,1]   is not a well ordered set as (0,1) is a subset of [0,1] and doesn't have a least element but if this is only taken for integers, then it is well ordered set.

Hence, the interval S = (0, 1) is not well-ordered if taken for real numbers

but if we take (0,1) for integers , it is well ordered .

Comments

..yes..u r right...but i m talking..abt integers..not real numbers..

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oh yess..oohk

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@Kapil . You have cited example of non-negative inetegers {0,1,2,3,4,5.......} . So, if I take non-empty subset {2,3,4......} then 2 will be least element, right?