0
$\begingroup$

Let < be a partial ordering on a nonempty set A. If every nonempty subset B of A has a <-least element, then < is a well ordering on A.

$\endgroup$
  • 1
    $\begingroup$ Is this a homework question? What have you tried so far? $\endgroup$ – EgoKilla May 8 '14 at 10:23
  • $\begingroup$ No, it is an exercise in a book. I think since < is a partial ordering on set A, so < is transitive on set A. Take any a,b in any subset B of A, a<b, a=b or b<a. Thus, < is a linear ordering on A. Since every subset A has a least element, then < is a well ordering on A. That was my thought. Is there any problem or mistake on it? $\endgroup$ – funny May 8 '14 at 10:31
2
$\begingroup$

If $<$ is a total ordering, then we are finished, because a well ordering is exactly a total ordering with the property that you have stated. But it is easy to see that it is a total ordering, because for any $a,b \in A$, the set $\{a,b\}$ has a minimal element, so either $a \leq b$ or $b \leq a$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.