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

1. 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!

2. Re: designing a set of axioms to guarantee the invertability of a discrete function

Originally Posted by jdodle7
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 $3.4\ne \left\lfloor {\dfrac{15.7}{4}} \right\rfloor =3$

3. 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.

4. Re: designing a set of axioms to guarantee the invertability of a discrete function

Originally Posted by jdodle7
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 $\displaystyle x = \left\lfloor {\frac{y}{z}} \right\rfloor$.

5. 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.