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Math Help - A well-ordered set

  1. #1
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    A well-ordered set

    If A is any well-ordered set of real numbers, then every subset of A has a least element. If B is a nonempty subset of A, then B has a least element.

    Question: Since A is well ordered and B has a least element, and since having a least element is not a sufficient condition for a nonempty set to be well-ordered, does it mean that B is not necessarily well-ordered?
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  2. #2
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    Quote Originally Posted by novice View Post
    If A is any well-ordered set of real numbers, then every subset of A has a least element. If B is a nonempty subset of A, then B has a least element.

    Question: Since A is well ordered and B has a least element, and since having a least element is not a sufficient condition for a nonempty set to be well-ordered, does it mean that B is not necessarily well-ordered?
    If B\subseteq A, is it true that each subset of B is a subset of A?
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  3. #3
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    Quote Originally Posted by Plato View Post
    If B\subseteq A, is it true that each subset of B is a subset of A?
    Beside it being a subset of A, what is the reason to believe the set B is well-ordered?
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  4. #4
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    Quote Originally Posted by Plato View Post
    If B\subseteq A, is it true that each subset of B is a subset of A?
    Plato,

    I honestly have spent the entire time, since your last reply, thinking of the question you put forth. I think I know what you were trying to show me, but I want to know if I understood it well.

    Here is the answer:

    Since the set A of real numbers is a well-ordered set, by definition, every subset of A must have a least element, and since B is a subset of A, it follows that every subset of B is a subset of A containing a least element. In turn, every subset of B has a least element. Consequently, by the same definition, B is a well-ordered set.

    I think you can be proud of me.
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  5. #5
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    Remember it is every nonempty subset...
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  6. #6
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    Thank you, Plato, I will remember not to leave out the word "non-empty,"

    Oh, I would have lost my case at court.
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