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Thread: A sets proof

  1. #1
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    A sets proof

    Hi again, another question I am having trouble starting:

    Let $\displaystyle I$ be a nonempty subset of $\displaystyle \mathbb{Z}$ such that:

    $\displaystyle (\forall x \in I)(\forall y \in I)[(x-y) \in I]$ and $\displaystyle (\forall z \in \mathbb{Z})(\forall x \in I)[z \cdot x \in I]$.

    Show that for some $\displaystyle n \in I, I = \{z \in \mathbb{Z} \colon z = xn \ for \ some \ x \in \mathbb{Z}\}$.

    Please no full solutions, but if someone could show me how to proceed
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  2. #2
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    Quote Originally Posted by nzmathman View Post
    Hi again, another question I am having trouble starting:

    Let $\displaystyle I$ be a nonempty subset of $\displaystyle \mathbb{Z}$ such that:

    $\displaystyle (\forall x \in I)(\forall y \in I)[(x-y) \in I]$ and $\displaystyle (\forall z \in \mathbb{Z})(\forall x \in I)[z \cdot x \in I]$.

    Show that for some $\displaystyle n \in I, I = \{z \in \mathbb{Z} \colon z = xn \ for \ some \ x \in \mathbb{Z}\}$.

    Please no full solutions, but if someone could show me how to proceed


    1) Take a minimal positive element in I (why is there such an element?)

    2) Applying Euclides algorithm show that any element in I is an integer multiple of the element you found in (1)

    3) Go grab a beer and be happy.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    1) Take a minimal positive element in I (why is there such an element?)

    2) Applying Euclides algorithm show that any element in I is an integer multiple of the element you found in (1)

    3) Go grab a beer and be happy.

    Tonio
    I know why a minimal element of I exists, but why can we conclude a minimal positive element exists?

    Also, how would I apply Euclidean algorithm to something this abstract?
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  4. #4
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    I know why a minimal element of I exists, but why can we conclude a minimal positive element exists?
    There is no minimal element of $\displaystyle I$. Take any $\displaystyle x\in I$. Then the set $\displaystyle \{zx\mid z\in\mathbb{Z}\}\subseteq I$ does not have a minimal element.

    To prove that there is a minimal positive element, it is sufficient to know that there is any positive element. This is natural numbers we are talking about

    Also, how would I apply Euclidean algorithm to something this abstract?
    Take any (w.l.o.g. positive) $\displaystyle x\in I$ and the minimal positive element $\displaystyle m$ of $\displaystyle I$. Then the GCD of $\displaystyle x$ and $\displaystyle m$ is a linear combination of $\displaystyle x$ and $\displaystyle m$ by Bézout's identity, an application of the Euclid's algorithm. From there it is easy to show that $\displaystyle m$ divides $\displaystyle x$.
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