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Math Help - Smallest Element proof

  1. #1
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    Smallest Element proof

    Hey all, need some help with the following proof:

    Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.

    All help appreciated!
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  2. #2
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    Re: Smallest Element proof

    Using what theorems? You can use either induction or the "well ordered" property of the positive integers to prove this. They are equivalent but I don't know which you have to use. In either case, I think I would start by looking at B= {a- b| a∈ A}, a set of positive integers.
    Last edited by HallsofIvy; January 15th 2012 at 11:32 AM.
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  3. #3
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    Re: Smallest Element proof

    Quote Originally Posted by jstarks44444 View Post
    Let A be a non-empty subset of Z and b ∈ Z, such that for each a ∈ A, b <= a. Then A has a smallest element.
    Isn't b the smallest element by definition?
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  4. #4
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    Re: Smallest Element proof

    this is supposed to use the Well-Ordering Principle as well as "construction," which I am not sure about
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  5. #5
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    Re: Smallest Element proof

    Ah, yes, b does not have to be an element of A. Then, as post #2 suggested, you can consider the set B = {a - b | a ∈ A}, which is a subset of natural numbers and thus has the smallest element.
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  6. #6
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    Re: Smallest Element proof

    Why can't 'b' be negative?
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  7. #7
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    Re: Smallest Element proof

    Quote Originally Posted by jstarks44444 View Post
    Why can't 'b' be negative?
    It can; this does not change anything.
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