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.

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    $\begingroup$ 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

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 $b \leq a$.


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