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Thread: designing a set of axioms to guarantee the invertability of a discrete function

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    designing a set of axioms to guarantee the invertability of a discrete function

    I need to guarantee the following statement will be true:
    $\displaystyle z = \lfloor\frac{y}{x}\rfloor \Rightarrow x = \lfloor\frac{y}{z}\rfloor $

    I can already guarantee that $\displaystyle 0 < x < 10^4*y $. One or both of the floor functions could also be changed to be ceiling functions if that will make the equality work.

    Thanks for your help!
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    Re: designing a set of axioms to guarantee the invertability of a discrete function

    Quote Originally Posted by jdodle7 View Post
    I need to guarantee the following statement will be true: $z = \lfloor\dfrac{y}{x}\rfloor \Rightarrow x = \lfloor\dfrac{y}{z}\rfloor $!
    I assume that you understand implications.
    $z = \left\lfloor {\dfrac{y}{x}} \right\rfloor $ means that $z$ is an integer.
    If $y=15.7~\&~x=3.4$ then $z=4$ BUT $3.4\ne \left\lfloor {\dfrac{15.7}{4}} \right\rfloor =3$
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    Re: designing a set of axioms to guarantee the invertability of a discrete function

    notice that the title of this thread implies that this statement alone is not going to be true. Which is why I need to design a set of axioms to guarantee that it is true. One such axiom that I probably should have stated originally is that $\displaystyle x$ is an integer.
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    Re: designing a set of axioms to guarantee the invertability of a discrete function

    Quote Originally Posted by jdodle7 View Post
    notice that the title of this thread implies that this statement alone is not going to be true. Which is why I need to design a set of axioms to guarantee that it is true. One such axiom that I probably should have stated originally is that $\displaystyle x$ is an integer.
    Frankly, I don't think that you know anything about automatics.
    That $x\in\mathbb{Z}$ may be a condition but it has nothing to do with the axioms.

    Now you may want to prove: If $x\in\mathbb{Z}$ and $\displaystyle z = \left\lfloor {\frac{y}{x}} \right\rfloor $ then $\displaystyle x = \left\lfloor {\frac{y}{z}} \right\rfloor $.
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    Re: designing a set of axioms to guarantee the invertability of a discrete function

    Here's one possibility:
    Axiom: there exist only a single number.

    Definition: we call that number "1".

    Since all numbers are equal to "1" your theorem follows trivially.
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