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Math Help - Well-ordered Relation Help!

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    Well-ordered Relation Help!

    Suppose every nonempty subset of a partially ordered set has a least element. Does it follow that this set is well-ordered? [A well-order relation is a linear order with the property that every non-empty subset S has a least element]
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    Quote Originally Posted by emlevy View Post
    Suppose every nonempty subset of a partially ordered set has a least element. Does it follow that this set is well-ordered? [A well-order relation is a linear order with the property that every non-empty subset S has a least element]
    If a & b are any two points in S, by the given the subset {a,b} has a least element.
    It is either a or b. So a<b or b<a.
    Therefore, S is fully ordered.
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    fully ordered?
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    Quote Originally Posted by emlevy View Post
    fully ordered?
    \text{linearly ordered }=\text{fully ordered }=\text{completely ordered }=\text{totally ordered}.
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